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Cm Math Dynamics

1 · Prerequisites

2 · Summary

The differential-dynamics layer of the non-relativistic classical mechanics collection, all of it mathematics: the local flow of an autonomous system and its dependence on initial data, the Poincaré return map, finite-time tangent growth rates, the hyperbolic exponential dichotomy with its Lyapunov stability and instability corollaries, the bilateral two-symbol shift as a chaotic map, the scope of Poincaré recurrence, and the quadratic normal-mode and energy-stability theory of a mechanical system in a chart. Every item on this page carries domain: mathematics and depends only on mathematical suppliers; nothing here asserts that any of these models describes nature, and the physical reading of forces, masses, energies and recurrences remains with the consuming classical-mechanics pages.

The flow layer begins with The local flow of an autonomous first-order system, which defines the maximal solutions of $x^{'} = F (x)$ , the flow domain $Ω$ and the partial maps $Φ_{t}$ . The local flow is a partial one-parameter group: identity, composition and inverse-time laws proves the identity, composition and inverse-time laws $Φ_{s + t} = Φ_{s} \circ Φ_{t}$ ; Continuous and Lipschitz dependence on the initial condition on a Picard-Lindelof cylinder gives the Lipschitz and Gronwall estimates on a Picard-Lindelöf cylinder; C¹ dependence of an autonomous flow on the initial condition, and the variational equation proves $D_{a} Φ_{t} (a) = Y_{a} (t)$ for the solution of the variational equation, and The flow domain of a C¹ autonomous field is open and the flow is C¹ on it shows that $Ω$ is open and $Φ$ of class $C^{1}$ on it, the tool used by A transversal return of a C¹ flow yields a local C¹ Poincaré map, with inverse given by the reversed crossing to construct a local $C^{1}$ Poincaré return map with inverse given by the reversed crossing; its first-return conclusion additionally assumes that the open reference segment avoids the closure of the section, and proves persistence of that exclusion. Finite-time tangent growth rates and Lyapunov exponents of a C¹ flow records $t^{- 1} \log (∥ D_{x} Φ_{t} (x) u ∥_{2} / ∥ u ∥_{2})$ and defines the Lyapunov exponent only for a forward-unbounded lifetime and an existing finite limit.

The symbolic layer is The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets, The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact and The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense, which together exhibit the full two-symbol shift as a continuous self-map of a sequentially compact metric space (compact under the Countable and Dependent Choice assumptions of the companion lemma, clause 4) that is topologically mixing, has dense periodic points and is sensitive with constant $1$ , in the sense fixed by Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list. This is a topological example, not a smoothly realized mechanical flow. Scope of Poincaré recurrence for mechanical systems: finite invariant volume and complete return maps collects the published recurrence statements with their hypotheses made explicit: finite invariant measure, a nonzero time map defined for all iterates, and almost-everywhere conclusions only.

The hyperbolicity layer is Exponential dichotomy for a hyperbolic endomorphism: invariant stable and unstable subspaces with exponential estimates, whose invariant splitting $V = E_{s} \oplus E_{u}$ with exponential estimates is obtained from the coprime factorisation of the minimal polynomial and the Jordan form, with no diagonalisability hypothesis. On it rest An equilibrium whose linearisation has all eigenvalues in the open left half-plane is locally exponentially stable, where a linearisation with all eigenvalues in the open left half-plane gives a locally exponentially stable equilibrium through the Lyapunov matrix $\int_{0}^{\infty} e^{t A^{T}} e^{t A} d t$ , and A hyperbolic equilibrium whose linearisation has an eigenvalue with positive real part is unstable, where an eigenvalue with positive real part gives nonlinear instability through an indefinite quadratic form. Both are nonlinear conclusions with complete conditional proofs.

The mechanical and modal layer is The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart (the field $G$ , its Lagrangian and energy, with an arbitrary $C^{2}$ positive definite kinetic metric) and Energy is conserved along solutions of the mechanical system with a configuration-dependent kinetic metric (exact energy conservation along solutions). The modal theory is Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity and Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities, which decouple $M q^{''} + K q = 0$ into $z_{i}^{''} + λ_{i} z_{i} = 0$ with $λ_{i} \geq 0$ for symmetric non-negative $K$ , including the free affine zero mode and the modal energies; A C² local minimum has vanishing gradient and positive semidefinite Hessian and A strict local minimum of the potential gives a Lyapunov stable equilibrium of the mechanical system, by energy conservation give Lyapunov stability at a strict minimum of the potential, and Linearisation of the mechanical system at an equilibrium: M(q_0) xi'' + D²U(q_0) xi = 0, and the small-oscillation quadratic model computes the linearised system $M (q_{0}) ξ^{''} + D^{2} U (q_{0}) ξ = 0$ . Scope of the quadratic-mode and energy-stability results: exact quadratic model versus linearisation approximation separates the exact quadratic model from the linearisation approximation.

The companion page cm-math-dynamics-examples records worked examples: the flow of a linear system, two coupled oscillators, the return map of the planar oscillator, finite checks in the shift, computed dichotomy constants, and a non-hyperbolic counterexample.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableprecheck passOpen item page →

The local flow of an autonomous first-order system

Definition

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, and let $F : U \to R^{n}$ be continuous and locally Lipschitz in the state variable in the sense of Local Lipschitz continuity in the state variable, locally uniform in time and parameters. Read the autonomous system

$x^{'} (t) = F (x (t))$

as the first-order system of First-order systems, initial value problems, and solutions on intervals with domain $D = R \times U \subseteq R \times R^{n}$ and right-hand side $(t, x) \mapsto F (x)$ . Its initial value problems satisfy the Picard-Lindelöf hypotheses, so by Every Picard–Lindelöf initial value problem has one maximal solution on an open interval every $a \in U$ has a unique maximal solution.

For $a \in U$ denote by

$x (\cdot; a) : I_{a} ⟶ U$

the unique maximal solution with $x (0; a) = a$ , where $I_{a} \subseteq R$ is its maximal interval of existence (Extensions, maximal solutions, maximal intervals, and global solutions); $I_{a}$ is an open interval containing $0$ , and every solution of $x^{'} = F (x)$ through $(0, a)$ is a restriction of $x (\cdot; a)$ .

The flow domain of $F$ is the set

$Ω : = {(t, a) \in R \times U : t \in I_{a}},$

and the local flow (also phase flow) of $F$ is the map

$Φ : Ω ⟶ U, Φ (t, a) : = x (t; a) .$

For $t \in R$ the time- $t$ map is the partial map

$Φ_{t} : Ω_{t} ⟶ U, Ω_{t} : = {a \in U : (t, a) \in Ω}, Φ_{t} (a) : = Φ (t, a) .$

Thus $Φ_{0} (a) = a$ whenever it is defined, and $(t, a) \in Ω$ exactly when the maximal solution of $x^{'} = F (x)$ through $a$ is defined at time $t$ .

Remarks

The word local refers to the domain: $Ω$ need not equal $R \times U$ . Whether $Ω = R \times U$ (completeness of the field) is a separate property of $F$ and is not asserted by this definition. Every statement about $Φ$ has to carry its domain condition $(t, a) \in Ω$ explicitly.
The local flow is the solution family of an autonomous field. It is a different object from the stroboscopic map of a time-periodic system (The stroboscopic (time- $P$ ) map of a $P$ -periodic first-order system), although for an autonomous field the stroboscopic map of period $P > 0$ is the restriction of $Φ_{P}$ to the set of initial states whose maximal solutions exist for a full period.
The identities $Φ_{0} (a) = a$ and $Φ_{t + s} (a) = Φ_{t} (Φ_{s} (a))$ are not part of the definition; they are proved from maximality and uniqueness in The local flow is a partial one-parameter group: identity, composition and inverse-time laws. The definition records only the assignment of solutions, so that no uniqueness statement is smuggled into a definition.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The local flow is a partial one-parameter group: identity, composition and inverse-time laws

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, let $F : U \to R^{n}$ be continuous and locally Lipschitz in the state variable (Local Lipschitz continuity in the state variable, locally uniform in time and parameters), and let $Φ : Ω \to U$ be the local flow of $x^{'} = F (x)$ as in The local flow of an autonomous first-order system, with maximal solutions $x (\cdot; a) : I_{a} \to U$ and flow domain $Ω = {(t, a) : t \in I_{a}}$ . Then:

$(0, a) \in Ω$ and $Φ (0, a) = a$ for every $a \in U$ ;
if $(s, a) \in Ω$ and $(t, Φ (s, a)) \in Ω$ , then $(s + t, a) \in Ω$ and $Φ (s + t, a) = Φ (t, Φ (s, a))$ ;
if $(t, a) \in Ω$ , then $(- t, Φ (t, a)) \in Ω$ and $Φ (- t, Φ (t, a)) = a$ .

Facts & Assumptions

Given: The open set $U$ , the field $F$ , and the local flow $Φ$ built from the unique maximal solutions $x (\cdot; a)$ through $(0, a)$ .

[F1]

For every $a \in U$ , $x (\cdot; a) : I_{a} \to U$ is a solution of $x^{'} = F (x)$ with $x (0; a) = a$ on the open interval $I_{a}$ containing $0$ , it is maximal, and every solution of that initial value problem is a restriction of it; moreover this maximal solution is unique (The local flow of an autonomous first-order system, Extensions, maximal solutions, maximal intervals, and global solutions, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, First-order systems, initial value problems, and solutions on intervals).

[F2]

If $y : J \to U$ is a solution of $x^{'} = F (x)$ and $s \in J$ , then the translate $y_{s} (σ) : = y (σ + s)$ , defined for $σ \in J - s : = {σ : σ + s \in J}$ , is again a solution of $x^{'} = F (x)$ on the open interval $J - s$ , with $y_{s} (0) = y (s)$ : this is the chain rule applied to the translation $σ \mapsto σ + s$ , which has derivative $1$ (First-order systems, initial value problems, and solutions on intervals).

Proof

technique · direct; use uniqueness and maximality of solutions to glue translated solutions

1.1givenF1

Clause 1 is definitional: $0 \in I_{a}$ because $I_{a}$ is the maximal interval of the solution through $(0, a)$ , and $Φ (0, a) = x (0; a) = a$ .

1.2givenF2

Fix $(s, a) \in Ω$ and put $b : = Φ (s, a) = x (s; a)$ . Define $J : = I_{a} - s$ and $y : J \to U$ by $y (σ) : = x (σ + s; a)$ . By [F2], $y$ solves $x^{'} = F (x)$ on the open interval $J$ , and $y (0) = x (s; a) = b$ .

2.1givenstep 1.2F1

The curve $x (\cdot; b)$ is the unique maximal solution through $(0, b)$ and $y$ is a solution through $(0, b)$ ; by [F1] $y$ is the restriction of $x (\cdot; b)$ to $J$ , so $J \subseteq I_{b}$ and $y = x (\cdot; b) ∣_{J}$ . If $J$ were a proper subinterval of $I_{b}$ , the curve $z (r) : = x (r - s; b)$ on $I_{b} + s$ would, by [F2], be a solution extending $x (\cdot; a)$ from $I_{a} = J + s$ to a strictly larger interval, contradicting its maximality in [F1]. Thus $I_{b} = J$ . Now let $t \in I_{b}$ ; then $t \in J$ , that is $s + t \in I_{a}$ , and $Φ (s + t, a) = x (s + t; a) = y (t) = x (t; b) = Φ (t, Φ (s, a))$ . This is clause 2.

3.1givenstep 2.1F1F2∎

For clause 3, fix $(t, a) \in Ω$ and put $b : = Φ (t, a) = x (t; a)$ . Define $K : = I_{a} - t$ and $w : K \to U$ by $w (σ) : = x (σ + t; a)$ . By [F2] (with $s = t$ ) $w$ solves $x^{'} = F (x)$ on the open interval $K$ , and $w (0) = x (t; a) = b$ . By [F1] $w$ is the restriction of the unique maximal solution $x (\cdot; b)$ to $K$ ; hence $- t \in K \subseteq I_{b}$ and $Φ (- t, b) = x (- t; b) = w (- t) = x (0; a) = a$ .

Remarks

Clause 2 is the cocycle or semigroup identity for the partial flow; it is proved, not assumed. Its hypothesis $(t, Φ (s, a)) \in Ω$ cannot be dropped: for a field whose solutions blow up in finite time, the solution through $a$ may be defined at time $s$ but not for a further time $t$ from $Φ (s, a)$ .
Clause 3 says that $Φ_{t}$ is injective on its domain with inverse $Φ_{- t}$ restricted to $Φ_{t} (Ω_{t})$ ; this is a consequence of uniqueness and maximality, and it is used in Finite-time tangent growth rates and Lyapunov exponents of a C¹ flow.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Continuous and Lipschitz dependence on the initial condition on a Picard-Lindelof cylinder

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open and let $F : U \to R^{n}$ be continuous and locally Lipschitz in the state variable (Local Lipschitz continuity in the state variable, locally uniform in time and parameters). Let $a_{0} \in U$ , let $h > 0$ and $R > 0$ satisfy $\overline{B} (a_{0}, R) \subseteq U$ (Open ball, closed ball and sphere in a metric space), put

$M : = \sup_{x \in \overline{B} (a_{0}, R)} ∥ F (x) ∥_{2}, L : = a state-Lipschitz constant of F on \overline{B} (a_{0}, R),$

so that $∥ F (x) - F (y) ∥_{2} \leq L ∥ x - y ∥_{2}$ for all $x, y \in \overline{B} (a_{0}, R)$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and let $0 \leq ρ < R$ satisfy

$h M \leq R - ρ, L h < 1.$

Then for every $a \in \overline{B} (a_{0}, ρ)$ the initial value problem

$x^{'} (t) = F (x (t)), x (0) = a$

has exactly one solution $x (\cdot; a)$ on $[- h, h]$ whose graph lies in $[- h, h] \times \overline{B} (a_{0}, R)$ , and for all $a, b \in \overline{B} (a_{0}, ρ)$ and all $s, t \in [- h, h]$ :

$∥ x (t; a) - x (t; b) ∥_{2} \leq e^{L ∣ t ∣} ∥ a - b ∥_{2}$ ;
$∥ x (t; a) - x (s; a) ∥_{2} \leq M ∣ t - s ∣$ ;
$∥ x (t; a) - x (s; b) ∥_{2} \leq e^{L h} ∥ a - b ∥_{2} + M ∣ t - s ∣$ ; in particular the map $(t, a) \mapsto x (t; a)$ is Lipschitz, hence continuous, on the compact set $[- h, h] \times \overline{B} (a_{0}, ρ)$ .

Facts & Assumptions

Given: The open set $U$ , the continuous and locally Lipschitz field $F$ , the point $a_{0} \in U$ , the numbers $h, R > 0$ with $\overline{B} (a_{0}, R) \subseteq U$ , the constants $M, L$ and the radius $ρ$ with $h M \leq R - ρ$ and $L h < 1$ .

[F1]

Picard-Lindelöf: if $[t_{0} - h, t_{0} + h] \times \overline{B} (x_{0}, r) \subseteq R \times U$ , if $∥ F ∥_{2} \leq M$ and $F$ has state-Lipschitz constant $L$ on that cylinder, and if $h M \leq r$ and $L h < 1$ , then exactly one solution of $x^{'} = F (x)$ , $x (t_{0}) = x_{0}$ exists on $[t_{0} - h, t_{0} + h]$ with graph in the cylinder; any two solutions through the same initial data agree on every common subinterval containing the initial time (Picard-Lindelöf local existence and uniqueness for first-order systems, First-order systems, initial value problems, and solutions on intervals).

[F2]

$F$ is locally Lipschitz in the state variable: every compact time-state cylinder carries a finite state-Lipschitz constant (Local Lipschitz continuity in the state variable, locally uniform in time and parameters); in particular $L$ as chosen in the statement is finite and the inequality $∥ F (x) - F (y) ∥_{2} \leq L ∥ x - y ∥_{2}$ holds on $\overline{B} (a_{0}, R)$ .

[F3]

For $ρ = 0$ the notation $\overline{B} (a_{0}, ρ)$ means the singleton ${a_{0}}$ , as follows from the inequality defining a closed ball. $\overline{B} (a_{0}, R)$ is closed and bounded, hence compact in $R^{n}$ (Open ball, closed ball and sphere in a metric space, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line), so the continuous function $∥ F ∥_{2}$ attains its maximum there and $M < + \infty$ (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[F4]

A curve solves the initial value problem on an interval exactly when it is continuous there, has graph in $U$ , and satisfies the Volterra equation $x (t) = a + \int_{0}^{t} F (x (r)) d r$ ; the oriented integral is additive over subintervals (A first-order initial value problem is equivalent to its Volterra integral equation, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[F5]

The Euclidean norm satisfies the triangle inequality (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation). For an integrable $R^{n}$ -valued curve $g$ on an interval and reals $r \leq s$ one has $∥ \int_{r}^{s} g ∥_{2} \leq ∣ \int_{r}^{s} ∥ g ∥_{2} ∣$ , and if a real integrable $u$ satisfies $u \leq c$ on a nondegenerate interval then $\int u \leq c (length)$ (For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable, If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[F6]

Gronwall: if continuous $u : [0, T] \to R$ and constants $A \geq 0$ , $L \geq 0$ satisfy $u (t) \leq A + L \int_{0}^{t} u$ for $t \in [0, T]$ , then $u (t) \leq A e^{L t}$ there; the reflected form for an integral oriented as $\int_{t}^{0}$ gives $u (t) \leq A e^{L ∣ t ∣}$ for $t \leq 0$ (Gronwall's integral inequality with variable and constant coefficients).

Proof

technique · direct; construct the solutions on a common cylinder and estimate their difference with Gronwall's inequality

1.1givenF1F2F3algebra

Fix $a \in \overline{B} (a_{0}, ρ)$ and put $r_{a} : = R - ∥ a - a_{0} ∥_{2}$ , so that $r_{a} \geq R - ρ > 0$ . Then $[- h, h] \times \overline{B} (a, r_{a}) \subseteq [- h, h] \times \overline{B} (a_{0}, R) \subseteq R \times U$ : if $∥ x - a ∥_{2} \leq r_{a}$ then $∥ x - a_{0} ∥_{2} \leq ∥ x - a ∥_{2} + ∥ a - a_{0} ∥_{2} \leq r_{a} + ∥ a - a_{0} ∥_{2} = R$ . Moreover $h M \leq R - ρ \leq r_{a}$ and $L h < 1$ , and the bounds $M$ and $L$ hold on the larger cylinder $[- h, h] \times \overline{B} (a_{0}, R)$ by [F2] and [F3]. By [F1] with $t_{0} = 0$ , $x_{0} = a$ , $r = r_{a}$ , exactly one solution $x (\cdot; a)$ on $[- h, h]$ exists whose graph lies in $[- h, h] \times \overline{B} (a, r_{a}) \subseteq [- h, h] \times \overline{B} (a_{0}, R)$ .

2.1F4F5step 1.1algebra

For $a, b \in \overline{B} (a_{0}, ρ)$ put $e : = x (\cdot; a) - x (\cdot; b)$ . By [F4], for $t \in [0, h]$ , $e (t) = a - b + \int_{0}^{t} (F (x (r; a)) - F (x (r; b))) d r .$ The integrand is continuous because $F$ and the two solutions are continuous, so [F5] gives $∥ e (t) ∥_{2} \leq ∥ a - b ∥_{2} + L \int_{0}^{t} ∥ e (r) ∥_{2} d r$ : the Lipschitz estimate applies on $\overline{B} (a_{0}, R)$ since both graphs lie there by step 1.1.

2.2step 1.1F4F5algebra

For $s, t \in [- h, h]$ , [F4] gives $x (t; a) - x (s; a) = \int_{s}^{t} F (x (r; a)) d r$ , and the graph of $x (\cdot; a)$ lies in $\overline{B} (a_{0}, R)$ by step 1.1, where $∥ F ∥ \leq M$ ; hence $∥ x (t; a) - x (s; a) ∥_{2} \leq M ∣ t - s ∣$ by [F5]. This is clause 2.

3.1step 2.1F1F4F5F6algebra

Applying [F6] with $u = ∥ e ∥_{2}$ , $A = ∥ a - b ∥_{2}$ gives $∥ e (t) ∥_{2} \leq e^{L t} ∥ a - b ∥_{2}$ for $t \in [0, h]$ . For $t \in [- h, 0]$ apply the same argument to the reflected curves $\tilde{x} (s) : = x (- s; a)$ , $\tilde{x} (s) : = x (- s; b)$ on $[0, - t]$ : their difference obeys the same integral inequality, so $∥ e (t) ∥_{2} \leq e^{- L t} ∥ a - b ∥_{2} = e^{L ∣ t ∣} ∥ a - b ∥_{2}$ . This is clause 1.

4.1step 3.12.2algebra∎

Combining clauses 1 and 2 with the triangle inequality, $∥ x (t; a) - x (s; b) ∥_{2} \leq ∥ x (t; a) - x (s; a) ∥_{2} + ∥ x (s; a) - x (s; b) ∥_{2} \leq M ∣ t - s ∣ + e^{L h} ∥ a - b ∥_{2}$ for all $(t, a), (s, b) \in [- h, h] \times \overline{B} (a_{0}, ρ)$ ; this is clause 3. The domain of the map is a closed bounded subset of $R \times R^{n}$ , hence compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line), and the displayed inequality exhibits it as Lipschitz.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

C¹ dependence of an autonomous flow on the initial condition, and the variational equation

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, let $F : U \to R^{n}$ be of class $C^{1}$ ( $C^{k}$ Euclidean maps and diffeomorphisms), let $a_{0} \in U$ , and let $T > 0$ , $R > 0$ with $\overline{B} (a_{0}, R) \subseteq U$ (Open ball, closed ball and sphere in a metric space). Put $M : = \sup_{\overline{B} (a_{0}, R)} ∥ F ∥_{2}$ and $L : = \sup_{\overline{B} (a_{0}, R)} ∥ D F ∥$ , where $D F (x)$ is the total derivative of $F$ at $x$ with its operator norm (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The operator norm as the least bound and as the unit-sphere or unit-ball supremum); both are finite. Let $0 < ρ < R - M T$ , so that in particular $M T < R$ . Let $Φ$ be the local flow of $x^{'} = F (x)$ (The local flow of an autonomous first-order system).

Then:

For every $a \in \overline{B} (a_{0}, ρ)$ the maximal solution $x (\cdot; a)$ is defined on $[0, T]$ and satisfies $∥ x (t; a) - a_{0} ∥_{2} \leq ρ + M t < R$ for $t \in [0, T]$ ; moreover

$∥ x (t; a) - x (t; b) ∥_{2} \leq e^{L t} ∥ a - b ∥_{2} (a, b \in \overline{B} (a_{0}, ρ), t \in [0, T]),$

and $(t, a) \mapsto x (t; a)$ is continuous on the compact set $[0, T] \times \overline{B} (a_{0}, ρ)$ .

For every $a \in B (a_{0}, ρ)$ the variational equation along $x (\cdot; a)$ ,

$Y^{'} (t) = D F (x (t; a)) Y (t), Y (0) = I_{n},$

has a unique solution $Y_{a} : [0, T] \to M_{n} (R)$ (The variational equation along an ODE solution), and for every $t \in [0, T]$ the map $a \mapsto x (t; a)$ is differentiable on the open ball $B (a_{0}, ρ)$ , with

$D_{a} x (t; a) h = Y_{a} (t) h (h \in R^{n}) .$

The map $(t, a) \mapsto Y_{a} (t)$ is continuous on $[0, T] \times B (a_{0}, ρ)$ , and consequently $(t, a) \mapsto x (t; a)$ is $C^{1}$ there with

$\partial_{t} x (t; a) = F (x (t; a)), D_{a} x (t; a) = Y_{a} (t) .$

Facts & Assumptions

Given: The open set $U$ , the $C^{1}$ field $F$ , the point $a_{0} \in U$ , the numbers $T, R > 0$ with $\overline{B} (a_{0}, R) \subseteq U$ , the finite constants $M, L$ , and $0 < ρ < R - M T$ .

[F10]

Matrix and norm convention (The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation, A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum): a matrix $A \in M_{n} (R)$ acts on Euclidean $R^{n}$ by $(A v)_{i} = \sum_{j < n} A_{i j} v_{j}$ , and every matrix norm in this item is the subordinate Euclidean norm $∥ A ∥_{o p} : = \sup_{∥ v ∥_{2} \leq 1} ∥ A v ∥_{2}$ ; the undecorated $∥ D F ∥$ in the Statement means this norm. Rowwise Cauchy-Schwarz gives $∥ A v ∥_{2} \leq (\sum_{i, j < n} A_{i j}^{2})^{1 / 2} ∥ v ∥_{2}$ , so the linear map is bounded and its operator norm is finite. Rescaling gives $∥ A v ∥_{2} \leq ∥ A ∥_{o p} ∥ v ∥_{2}$ ; the triangle inequality on vectors gives the triangle inequality on these operator norms. Matrix multiplication represents composition, hence $∥ A B ∥_{o p} \leq ∥ A ∥_{o p} ∥ B ∥_{o p}$ , and $∥ I_{n} ∥_{o p} = 1$ since $n \geq 1$ . Moreover $∣ ∥ A ∥_{o p} - ∥ B ∥_{o p} ∣ \leq ∥ A - B ∥_{o p} \leq (\sum_{i, j < n} (A_{i j} - B_{i j})^{2})^{1 / 2}$ , so entrywise continuity implies continuity in this norm. For a continuous matrix curve $C$ on $[r, s]$ , its integral is entrywise; applying the vector integral estimate to $C (t) v$ for each $∥ v ∥_{2} \leq 1$ and taking the supremum gives $∥ \int_{r}^{s} C (t) d t ∥_{o p} \leq \int_{r}^{s} ∥ C (t) ∥_{o p} d t$ when $r \leq s$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable).

[F1]

$F$ is continuous (differentiability implies continuity) and locally Lipschitz in the state variable: on the compact convex cylinder $[0, T] \times \overline{B} (a_{0}, R)$ the derivative bound $L$ gives $∥ F (x) - F (y) ∥_{2} \leq L ∥ x - y ∥_{2}$ , since $F$ is $C^{1}$ and the segment between two points of the ball lies in the ball; and $∥ F ∥ \leq M$ there. The same segment argument on a closed ball contained in $U$ about any point proves the local Lipschitz hypothesis throughout $U$ . Also $\overline{B} (a_{0}, R)$ is compact, and [F10] makes $∥ D F ∥_{o p}$ continuous, so both suprema are attained and finite (Local Lipschitz continuity in the state variable, locally uniform in time and parameters, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F2]

Picard-Lindelöf and maximal solutions: local solutions exist uniquely through every initial point, and every initial value problem has a unique maximal solution, of which every other solution through the same data is a restriction (Picard-Lindelöf local existence and uniqueness for first-order systems, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, Extensions, maximal solutions, maximal intervals, and global solutions, The local flow of an autonomous first-order system).

[F3]

Compact continuation: a solution of $x^{'} = F (x)$ whose graph approaches a compact subset of $R \times U$ at a finite endpoint extends past that endpoint; in the autonomous case it suffices that the solution values stay in a compact subset of $U$ near the endpoint (Compact continuation for locally Lipschitz ODEs: a solution whose graph stays in a compact subset of the domain near a finite endpoint extends past it).

[F4]

Solutions are equivalent to solutions of the Volterra equation $x (t) = a + \int_{0}^{t} F (x (r)) d r$ , and the integral is additive over subintervals (A first-order initial value problem is equivalent to its Volterra integral equation, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[F5]

Norm and integral estimates: $∥ \int_{r}^{s} g ∥_{2} \leq ∣ \int_{r}^{s} ∥ g ∥_{2} ∣$ , and if $∥ g ∥_{2} \leq c$ on an interval then the norm of the integral is at most $c$ times the length; the Euclidean norm satisfies the triangle inequality, and for $u \neq 0$ the supremum $\sup_{∥ v ∥_{2} = 1} ∣ ⟨ u, v ⟩_{2} ∣$ equals $∥ u ∥_{2}$ (the value $v = u / ∥ u ∥_{2}$ gives $∥ u ∥_{2}$ , and Cauchy-Schwarz bounds every value by $∥ u ∥_{2}$ ) (For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable, If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F6]

Gronwall: if continuous $u : [0, T] \to R$ , $u \geq 0$ , satisfies $u (t) \leq a (t) + \int_{0}^{t} L u$ with $L \geq 0$ and nondecreasing continuous $a \geq 0$ , then $u (t) \leq a (t) e^{L t}$ ; with a constant $A$ this reads $u (t) \leq A e^{L t}$ (Gronwall's integral inequality with variable and constant coefficients).

[F7]

$F$ is $C^{1}$ , so for $x, y$ with the segment $[x, y] \subseteq U$ one has $F (y) - F (x) = \int_{0}^{1} D F (x + τ (y - x)) (y - x) d τ$ , by the chain rule for $τ \mapsto F (x + τ (y - x))$ and the fundamental theorem; on the compact ball this gives the uniform remainder bound $∥ F (y) - F (x) - D F (x) (y - x) ∥_{2} \leq ω (∥ y - x ∥_{2}) ∥ y - x ∥_{2}$ , where $ω (s) : = \sup {∥ D F (u) - D F (w) ∥ : u, w \in \overline{B} (a_{0}, R), ∥ u - w ∥_{2} \leq s}$ and $ω (s) \to 0$ as $s ↓ 0$ by [F10] and Heine-Cantor uniform continuity of $D F$ on the compact ball (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[F8]

The linear matrix initial value problem $Y^{'} = C (t) Y$ , $Y (0) = I_{n}$ , with $C : [0, T] \to M_{n} (R)$ continuous, has a unique solution on all of $[0, T]$ (Linear matrix ODEs have unique global solutions on a fixed interval).

[F9]

If the partial derivatives of a map exist on a neighbourhood of a point and are continuous there, the map is totally differentiable at that point (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

Proof

technique · direct; first trap the solutions in a compact ball and extend them by continuation, then estimate differences with Gronwall, then compare solutions with the variational equation

1.1givenF1F2F4F5choosealgebra

Fix $a \in \overline{B} (a_{0}, ρ)$ and let $x = x (\cdot; a) : [0, β) \to U$ be the maximal forward solution, with $β \leq \infty$ . Choose a fixed radius $r$ with $ρ + M T < r < R$ . The curve cannot reach the sphere of radius $r$ about $a_{0}$ at a time $t_{*} < β$ with $t_{*} \leq T$ : at the first such time it has stayed in $\overline{B} (a_{0}, r) \subseteq \overline{B} (a_{0}, R)$ , so [F4] and [F5] give $∥ x (t_{*}) - a_{0} ∥_{2} \leq ρ + M t_{*} \leq ρ + M T < r$ , a contradiction. Thus for every $0 \leq t < β$ with $t \leq T$ the whole curve on $[0, t]$ stays in this fixed ball, and the same integral estimate gives $∥ x (t) - a_{0} ∥_{2} \leq ρ + M t < R$ .

2.1step 1.1F3

Consequently, if $β \leq T$ the graph of $x$ stays in the compact set $[0, T] \times \overline{B} (a_{0}, ρ + M T) \subseteq R \times U$ ; by [F3] $x$ extends past $β$ , contradicting maximality of the maximal solution (Extensions, maximal solutions, maximal intervals, and global solutions). Hence $β > T$ , so $x (\cdot; a)$ is defined on $[0, T]$ , and step 1.1 gives $∥ x (t; a) - a_{0} ∥_{2} \leq ρ + M t \leq ρ + M T < R$ there.

3.1step 1.12.1F1F4F5F6algebra

For $a, b \in \overline{B} (a_{0}, ρ)$ put $e (t) : = x (t; a) - x (t; b)$ . By [F4] and [F5], $∥ e (t) ∥_{2} \leq ∥ a - b ∥_{2} + L \int_{0}^{t} ∥ e (r) ∥_{2} d r$ for $t \in [0, T]$ , because both graphs lie in $\overline{B} (a_{0}, R)$ by step 2.1 and $∥ F (x) - F (y) ∥ \leq L ∥ x - y ∥$ there. By [F6] with $A = ∥ a - b ∥_{2}$ , $∥ e (t) ∥_{2} \leq e^{L t} ∥ a - b ∥_{2}$ . Adding the uniform bound $∥ x (t; a) - x (s; a) ∥_{2} \leq M ∣ t - s ∣$ obtained from [F4] and [F5] gives continuity of $(t, a) \mapsto x (t; a)$ on $[0, T] \times \overline{B} (a_{0}, ρ)$ . This is clause 1.

4.1step 3.1F7F8algebra

Fix $a \in B (a_{0}, ρ)$ and let $Y_{a}$ be the unique solution of the variational equation on $[0, T]$ , which exists by [F8] since $t \mapsto D F (x (t; a))$ is continuous (composition of the continuous map $D F$ with the continuous curve $x (\cdot; a)$ ). For $h \in R^{n}$ with $a + h \in \overline{B} (a_{0}, ρ)$ put $e_{h} : = x (\cdot; a + h) - x (\cdot; a)$ and $R_{h} (t) : = F (x (t; a + h)) - F (x (t; a)) - D F (x (t; a)) e_{h} (t) .$ By [F7] applied to the pair $(x (t; a + h), x (t; a))$ , whose segment lies in $\overline{B} (a_{0}, R)$ by convexity, $∥ R_{h} (t) ∥_{2} \leq ω (∥ e_{h} (t) ∥_{2}) ∥ e_{h} (t) ∥_{2}$ , and by step 3.1 $∥ e_{h} (t) ∥_{2} \leq e^{L T} ∥ h ∥_{2}$ .

4.2step 3.1F4F5F6F7F10algebra

The map $(t, a) \mapsto Y_{a} (t)$ is continuous on $[0, T] \times B (a_{0}, ρ)$ . First, $∥ Y_{a} (t) ∥_{o p} \leq e^{L t} \leq e^{L T}$ by the entrywise Volterra equation, the matrix integral and product estimates of [F10], $∥ I_{n} ∥_{o p} = 1$ and [F6]. Second, for $a, b$ the difference $Δ : = Y_{a} - Y_{b}$ satisfies $Δ^{'} = D F (x (t; a)) Δ + (D F (x (t; a)) - D F (x (t; b))) Y_{b}$ with $Δ (0) = 0$ , so [F4], [F5], [F6] and [F10] with $∥ Y_{b} ∥_{o p} \leq e^{L T}$ give $∥ Δ (t) ∥_{o p} \leq e^{L T} \cdot T \cdot \sup_{r \in [0, T]} ∥ D F (x (r; a)) - D F (x (r; b)) ∥_{o p} \cdot e^{L T},$ which tends to $0$ as $b \to a$ because $x (r; b) \to x (r; a)$ uniformly in $r$ by step 3.1 and $∥ D F (u) - D F (w) ∥ \leq ω (∥ u - w ∥_{2})$ with $ω (s) \to 0$ . Finally, $∥ Y_{a} (t) - Y_{a} (s) ∥_{o p} \leq L e^{L T} ∣ t - s ∣$ by $Y_{a} (t) - Y_{a} (s) = \int_{s}^{t} D F (x (r; a)) Y_{a} (r) d r$ and [F10]. Combining these three estimates proves continuity.

5.1step 4.1F4F5F6F7algebra

Define $δ_{h} (t) : = e_{h} (t) - Y_{a} (t) h$ . Differentiating and using the definitions of $e_{h}$ , $R_{h}$ and $Y_{a}$ gives $δ_{h}^{'} (t) = D F (x (t; a)) δ_{h} (t) + R_{h} (t)$ with $δ_{h} (0) = 0$ ; hence, by [F4] applied to the curve $δ_{h}$ and [F5], $∥ δ_{h} (t) ∥_{2} \leq \int_{0}^{t} ∥ R_{h} (r) ∥_{2} d r + L \int_{0}^{t} ∥ δ_{h} (r) ∥_{2} d r .$ Since $r \mapsto \int_{0}^{r} ∥ R_{h} ∥_{2}$ is nondecreasing, [F6] gives $∥ δ_{h} (t) ∥_{2} \leq e^{L t} \int_{0}^{t} ∥ R_{h} (r) ∥_{2} d r \leq e^{2 L T} T ω (e^{L T} ∥ h ∥_{2}) ∥ h ∥_{2}$ for $∥ h ∥_{2} \leq e^{- L T}$ ; dividing by $∥ h ∥_{2}$ and letting $h \to 0$ yields $∥ x (t; a + h) - x (t; a) - Y_{a} (t) h ∥_{2} / ∥ h ∥_{2} \to 0$ uniformly in $t \in [0, T]$ . Hence $a \mapsto x (t; a)$ is differentiable on $B (a_{0}, ρ)$ with $D_{a} x (t; a) h = Y_{a} (t) h$ for every $t$ . This is clause 2.

6.1step 1.13.14.25.1F2F8F9choose∎

Choose $ϵ > 0$ so small that $ρ + M (T + ϵ) < R$ (if $M = 0$ , any sufficiently small positive $ϵ$ works). Steps 1.1-5.1, which did not use [F9], apply with $T + ϵ$ in place of $T$ and show that the positive-time flow has continuous state derivative on $[0, T + ϵ] \times B (a_{0}, ρ)$ . Apply those same steps to the $C^{1}$ field $- F$ for time length $ϵ$ , with the same ball and bounds $M, L$ . Its solution $Ψ (s, a)$ gives $Φ (t, a) = Ψ (- t, a)$ for $- ϵ \leq t \leq 0$ : time reversal solves the original equation and agrees with its maximal solution by [F2]. Hence the state derivative exists and is continuous also on this negative-time slab; at $t = 0$ both state derivatives are $I_{n}$ , and their joint continuity on the two slabs makes the state derivative continuous across $0$ . The time derivative exists throughout the open set $(- ϵ, T + ϵ) \times B (a_{0}, ρ)$ and equals $F (Φ (t, a))$ , including at $0$ because the maximal solution solves the equation on its open interval. Joint continuity follows from step 3.1 on each slab and their common value $Φ (0, a) = a$ . Thus all partial derivatives are continuous on this open set, and [F9] makes the flow $C^{1}$ there. Its restriction proves clause 3 on $[0, T] \times B (a_{0}, ρ)$ , with $D_{a} Φ (t, a) = Y_{a} (t)$ by step 5.1; the variational solution on the longer interval restricts to $Y_{a}$ by uniqueness [F8].

Remarks

The hypothesis $ρ + M T < R$ is a genuine trapping condition: it keeps the whole graph of every considered solution in a compact subset of the domain, which is what lets Compact continuation for locally Lipschitz ODEs: a solution whose graph stays in a compact subset of the domain near a finite endpoint extends past it extend the maximal solution to the full interval $[0, T]$ . The conclusion is local in the initial condition and in time, and no claim is made that $D_{a} x (t; a)$ is defined at the boundary of the ball or for $t$ outside $[0, T]$ .
The proof actually shows more than stated: $∥ x (t; a + h) - x (t; a) - Y_{a} (t) h ∥_{2} \leq e^{2 L T} T ω (e^{L T} ∥ h ∥_{2}) ∥ h ∥_{2}$ , an explicit first-order error bound. The nonautonomous analogue (with $F = F (t, x)$ and the variational equation along a solution on a compact interval) is the same computation with the time variable kept, and is not needed in this batch.

CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The flow domain of a C¹ autonomous field is open and the flow is C¹ on it

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, let $F : U \to R^{n}$ be of class $C^{1}$ ( $C^{k}$ Euclidean maps and diffeomorphisms), and let $Φ : Ω \to U$ be the local flow of $x^{'} = F (x)$ with maximal solutions $x (\cdot; a) : I_{a} \to U$ (The local flow of an autonomous first-order system). Then $Ω$ is open in $R \times U$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement), and $Φ$ is of class $C^{1}$ on $Ω$ , with

$\partial_{t} Φ (t, a) = F (Φ (t, a)) ((t, a) \in Ω) .$

Facts & Assumptions

Given: The open set $U$ , the $C^{1}$ field $F$ and its local flow $Φ$ .

[F1]

Throughout this proof, vector norms are Euclidean and matrix norms are the subordinate Euclidean operator norms defined, bounded and shown continuous in Facts [F10] of C¹ dependence of an autonomous flow on the initial condition, and the variational equation; in particular $∥ D F ∥$ below means $∥ D F ∥_{o p}$ . Smooth dependence on the initial condition: if $a_{0} \in U$ , $T > 0$ , $R > 0$ with $\overline{B} (a_{0}, R) \subseteq U$ , $M = \sup_{\overline{B} (a_{0}, R)} ∥ F ∥_{2}$ , $L = \sup_{\overline{B} (a_{0}, R)} ∥ D F ∥$ and $0 < ρ < R - M T$ , then every $a \in \overline{B} (a_{0}, ρ)$ has its maximal solution defined on $[0, T]$ with values in $\overline{B} (a_{0}, ρ + M T) \subseteq B (a_{0}, R)$ , the estimate $∥ x (t; a) - x (t; b) ∥_{2} \leq e^{L t} ∥ a - b ∥_{2}$ holds for $t \in [0, T]$ , and $(t, a) \mapsto x (t, a)$ is $C^{1}$ on $[0, T] \times B (a_{0}, ρ)$ with $\partial_{t} x = F (x)$ and $D_{a} x (t; a) = Y_{a} (t)$ (C¹ dependence of an autonomous flow on the initial condition, and the variational equation).

[F2]

Composition law: $Φ (0, a) = a$ , and $Φ (s + t, a) = Φ (t, Φ (s, a))$ whenever both sides are defined (The local flow is a partial one-parameter group: identity, composition and inverse-time laws); every solution of $x^{'} = F (x)$ through an initial point is a restriction of the corresponding maximal solution (The local flow of an autonomous first-order system).

[F3]

Positive distance gap: if $K \subseteq R^{n}$ is nonempty compact, $C \subseteq R^{n}$ nonempty closed and $K \cap C = \emptyset$ , there is $δ > 0$ with $∥ k - c ∥_{2} \geq δ$ for all $k \in K$ , $c \in C$ (A compact set and a disjoint closed set have a positive norm-distance gap).

[F4]

Continuous images of compact sets are compact (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset); a continuous function on a nonempty compact set attains finite extrema (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value); closed bounded subsets of $R^{n}$ are compact and compact subsets are closed and bounded (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, A compact subset of a metric space is closed and bounded).

[F5]

Chain rule and $C^{1}$ calculus: the composite of totally differentiable maps is totally differentiable with $D (g \circ f) (x) = D g (f (x)) \circ D f (x)$ ; a map with continuous partial derivatives is totally differentiable; and a map that is totally differentiable everywhere with continuous total derivative is $C^{1}$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, $C^{k}$ Euclidean maps and diffeomorphisms, Open ball, closed ball and sphere in a metric space).

Proof

technique · direct; cover the reference trajectory by finitely many Picard-Lindelöf steps with radially shrinking balls and invoke the smooth dependence theorem on each step

1.1givenF1F3F4choosealgebra

Let $(t_{0}, a_{0}) \in Ω$ with $t_{0} > 0$ , and let $x : = x (\cdot; a_{0})$ on $[0, t_{0}]$ . Its image $K : = x ([0, t_{0}])$ is nonempty compact and lies in $U$ . If $R^{n} ∖ U$ is nonempty, [F3] gives a separation $d > 0$ ; choose $0 < δ < d$ , so every closed $δ$ -ball centred in $K$ lies in $U$ . If $U = R^{n}$ , choose any $δ > 0$ . Define $d_{K} (z) : = \min_{k \in K} ∥ z - k ∥_{2}$ : for each fixed $z$ , the reverse triangle inequality makes $k \mapsto ∥ z - k ∥_{2}$ continuous, so the minimum is attained by [F4]. If $k_{w}$ attains the minimum for $w$ , then $d_{K} (z) \leq ∥ z - k_{w} ∥_{2} \leq ∥ z - w ∥_{2} + d_{K} (w)$ ; interchanging $z, w$ gives $∣ d_{K} (z) - d_{K} (w) ∣ \leq ∥ z - w ∥_{2}$ . Thus the fixed tube $H : = {z : d_{K} (z) \leq δ}$ is closed. Since compact $K$ is bounded, there is $B \geq 0$ with $∥ k ∥_{2} \leq B$ for $k \in K$ ; a minimizing $k$ for $z \in H$ gives $∥ z ∥_{2} \leq B + δ$ , so $H$ is bounded and hence compact by [F4]. It lies in $U$ : if the complement of $U$ is nonempty, every point in it has $d_{K} (z) \geq d > δ$ by [F3]; when $U = R^{n}$ inclusion is automatic. The vector norm inequalities used here are those of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation. Put $M : = \sup_{H} ∥ F ∥_{2}$ and $L : = \sup_{H} ∥ D F ∥$ , finite by [F4] and the norm continuity in [F1], before choosing any subdivision. Choose $N \geq 1$ with $h : = t_{0} / N$ satisfying $h M \leq δ / 4$ and $h L \leq 1 / 2$ , and set $x_{j} : = x (j h)$ for $0 \leq j \leq N$ . Every $\overline{B} (x_{j}, δ)$ lies in $H$ ; thus its bounds $M_{j}, L_{j}$ satisfy $M_{j} \leq M$ , $L_{j} \leq L$ . Set $ρ : = δ / 4$ .

2.1step 1.1F1algebra

Small-step claim: for every $j$ and every $y \in \overline{B} (x_{j}, ρ)$ , the maximal solution through $y$ is defined on $[0, h]$ with values in $\overline{B} (x_{j}, δ) \subseteq U$ , and $Φ$ is $C^{1}$ on $[0, h] \times B (x_{j}, ρ)$ with $∥ Φ (t, y) - Φ (t, x_{j}) ∥_{2} \leq e^{L t} ∥ y - x_{j} ∥_{2}$ . Indeed $ρ + M_{j} h \leq δ / 4 + δ / 4 < δ$ , so [F1] applies to $a_{0} : = x_{j}$ , $T : = h$ , $R : = δ$ and the radius $ρ$ : the bound on $[0, h]$ is $M_{j} \leq M$ , and the stated estimates are clauses 1 and 3 of [F1].

3.1step 2.1F2F5algebra

Put $ρ_{0} : = ρ e^{- L t_{0}}$ and, for $0 \leq j \leq N$ , $ρ_{j} : = ρ e^{- L (t_{0} - j h)}$ , so that $ρ_{N} = ρ$ , $ρ_{j} \leq ρ$ for $j < N$ , and $ρ_{j + 1} = e^{L h} ρ_{j}$ . I claim, by induction on $j$ , that $Φ (j h, \cdot)$ is $C^{1}$ on $W : = B (a_{0}, ρ_{0})$ and $Φ (j h, W) \subseteq B (x_{j}, ρ_{j})$ . For $j = 0$ this says $Φ (0, a) = a \in B (a_{0}, ρ_{0})$ and is [F2]. If it holds for $j$ , then for $a \in W$ the point $y : = Φ (j h, a)$ lies in $B (x_{j}, ρ_{j}) \subseteq B (x_{j}, ρ)$ , so the small-step claim of step 2.1 gives that $Φ (h, y)$ is defined and $Φ ((j + 1) h, a) = Φ (h, Φ (j h, a))$ ; moreover $∥ Φ (h, y) - x_{j + 1} ∥_{2} = ∥ Φ (h, y) - Φ (h, x_{j}) ∥_{2} \leq e^{L h} ∥ y - x_{j} ∥_{2} < e^{L h} ρ_{j} = ρ_{j + 1}$ , so $Φ ((j + 1) h, W) \subseteq B (x_{j + 1}, ρ_{j + 1})$ , and $Φ ((j + 1) h, \cdot)$ is a composite of $C^{1}$ maps on the relevant open sets by [F5]. This proves the claim; in particular $Φ (t_{0}, \cdot)$ is $C^{1}$ on $W$ .

4.1step 2.13.1F1F2F5

The endpoint regularity in [F1] means regularity of a local extension to an open time neighbourhood, as its proof establishes. For $(t_{0}, a_{0})$ choose the last step, writing $t_{0} = (N - 1) h + h$ . By step 3.1, $g (a) : = Φ ((N - 1) h, a)$ is $C^{1}$ on $W$ and $g (a_{0}) = x_{N - 1}$ . By step 2.1 and [F1], the small-step flow is defined and $C^{1}$ on an open neighbourhood $V$ of $(h, x_{N - 1})$ . Continuity of $(t, a) \mapsto (t - (N - 1) h, g (a))$ gives a product neighbourhood $J \times W^{'}$ of $(t_{0}, a_{0})$ , with $W^{'} \subseteq W$ , whose image lies in $V$ . On it [F2] gives $Φ (t, a) = Φ (t - (N - 1) h, g (a))$ , so the flow is defined and $C^{1}$ by [F5]. This proves openness and regularity near every positive-time point without requiring any seam assertion.

5.1step 1.12.13.14.1F1F2F5choose∎

For a point $(0, a_{0})$ choose $R > 0$ with $\overline{B} (a_{0}, R) \subseteq U$ , put $M : = \sup_{\overline{B} (a_{0}, R)} ∥ F ∥_{2}$ , and choose $ρ = R / 2$ and $h > 0$ with $M h < R / 2$ . Apply [F1] to both $F$ and $- F$ on $[0, h] \times B (a_{0}, ρ)$ . Time reversal and uniqueness [F2] identify the second flow with $Φ (- s, a)$ , so $(- h, h) \times B (a_{0}, ρ) \subseteq Ω$ . The state derivatives are continuous on both sides and equal $I_{n}$ at $0$ , while the time derivative on the whole interval is $F (Φ (t, a))$ by the maximal-solution equation. The two continuous pieces agree at $0$ , so [F5] gives $C^{1}$ regularity on this open product. For $t_{0} < 0$ , apply steps 1.1-4.1 to $- F$ at $(- t_{0}, a_{0})$ and reverse time; uniqueness identifies its maximal interval with $- I_{a}$ (a proper extension would reverse to an extension of the original maximal solution). Thus every point of $Ω$ has an open neighbourhood on which $Φ$ is defined and $C^{1}$ . On all these neighbourhoods $\partial_{t} Φ (t, a) = F (Φ (t, a))$ , proving the statement.

Remarks

The proof uses only finitely many Picard-Lindelöf steps, so no countable or dependent choice is invoked; the positive gap of [F3] is the published choice-free separation statement. The corollary is the tool that lets the Poincaré return map of A transversal return of a C¹ flow yields a local C¹ Poincaré map, with inverse given by the reversed crossing be built from the implicit function theorem without extra hypotheses on the flow.
The conclusion is local in nature: $Ω$ is open but need not be all of $R \times U$ , and the corollary does not assert completeness of $F$ , boundedness of $Φ$ , or any uniformity of the $C^{1}$ estimate over all of $Ω$ .

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A transversal return of a C¹ flow yields a local C¹ Poincaré map, with inverse given by the reversed crossing

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, let $F : U \to R^{n}$ be of class $C^{1}$ ( $C^{k}$ Euclidean maps and diffeomorphisms), and let $Φ : Ω \to U$ be the local flow of $x^{'} = F (x)$ (The local flow of an autonomous first-order system). Let $x_{0} \in U$ and $t_{0} > 0$ satisfy $(t_{0}, x_{0}) \in Ω$ , and put $x_{1} : = Φ (t_{0}, x_{0})$ . Let $W \subseteq R^{n}$ be open with $x_{0}, x_{1} \in W$ , and let $h : W \to R$ be $C^{1}$ with

$h (x_{0}) = h (x_{1}) = 0, D h (x_{0}) F (x_{0}) \neq 0, D h (x_{1}) F (x_{1}) \neq 0;$

thus the reference orbit meets the level set $Σ : = h^{- 1} (0) \cap W$ transversally at $x_{0}$ (time $0$ ) and again at $x_{1}$ (time $t_{0}$ ). Then there are open sets $A \subseteq W$ with $x_{0} \in A$ and $B \subseteq W$ with $x_{1} \in B$ , and $C^{1}$ functions

$τ : A \to R, σ : B \to R, τ (x_{0}) = σ (x_{1}) = t_{0},$

such that:

$h (Φ (τ (x), x)) = 0$ for every $x \in A$ , and there is an open interval $J ∋ t_{0}$ with $A \subseteq W$ such that

$h (Φ (t, x)) = 0, (t, x) \in J \times A ⟺ t = τ (x);$

$h (Φ (- σ (y), y)) = 0$ for every $y \in B$ , and the map

$P : A ⟶ Σ, P (x) : = Φ (τ (x), x),$

is $C^{1}$ with $P (A \cap Σ) \subseteq B$ , while

$Q : B ⟶ Σ, Q (y) : = Φ (- σ (y), y),$

is $C^{1}$ with $Q (B \cap Σ) \subseteq A$ , $Q (P (x)) = x$ for all $x \in A \cap Σ$ , and $P (Q (y)) = y$ for all $y \in B \cap Σ \cap Q^{- 1} (A)$ ;

$P (A \cap Σ) = B \cap Σ \cap Q^{- 1} (A)$ is open in the relative topology of $Σ$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace), and $P : A \cap Σ \to P (A \cap Σ)$ is a $C^{1}$ diffeomorphism with inverse $Q ∣_{P (A \cap Σ)}$ . The map $P$ is the local Poincaré return map associated with the crossing at $t_{0}$ . Here the portions of $Σ$ in $A$ and $B$ have the embedded $C^{1}$ section charts supplied below; no regularity of $Σ$ away from these neighbourhoods is required.
If, in addition, $Φ (s, x_{0}) \notin {\overline{Σ}}^{U}$ for every $0 < s < t_{0}$ , where the bar denotes closure in $U$ (Interior, closure, boundary, exterior, derived set and isolated point in a topological space), then $A, B$ can be chosen so that $τ (x) > 0$ and $Φ (s, x) \notin Σ$ for $x \in A \cap Σ$ and $0 < s < τ (x)$ . In this case $P$ is the local first-return map to $Σ$ : $τ (x)$ is its least positive hitting time. This explicit exclusion condition supplies persistence even when $Σ$ is not closed in $U$ .

Facts & Assumptions

Given: The open set $U$ , the $C^{1}$ field $F$ , its local flow $Φ$ , the reference data $(t_{0}, x_{0})$ with $x_{1} = Φ (t_{0}, x_{0})$ , and the transversal function $h$ with the displayed properties.

[F1]

$Ω$ is open in $R \times U$ and $Φ : Ω \to U$ is $C^{1}$ , with $\partial_{t} Φ (t, a) = F (Φ (t, a))$ (The flow domain of a C¹ autonomous field is open and the flow is C¹ on it).

[F2]

Composition law and inverse time: $Φ (0, a) = a$ ; if $(s, a) \in Ω$ and $(t, Φ (s, a)) \in Ω$ then $(s + t, a) \in Ω$ and $Φ (s + t, a) = Φ (t, Φ (s, a))$ ; and $(t, a) \in Ω$ implies $(- t, Φ (t, a)) \in Ω$ with $Φ (- t, Φ (t, a)) = a$ (The local flow is a partial one-parameter group: identity, composition and inverse-time laws).

[F3]

Implicit function theorem: if $G$ is $C^{1}$ on an open set, $G (t_{0}, x_{0}) = 0$ and the partial derivative in the scalar variable at $(t_{0}, x_{0})$ is nonzero, then near $(t_{0}, x_{0})$ the zero set of $G$ is the graph of a unique $C^{1}$ function of the remaining variables, with the stated uniqueness for all points of a product neighbourhood (The Euclidean implicit function theorem with derivative formula).

[F4]

The chain rule gives $\partial_{t} (h (Φ (t, x))) = D h (Φ (t, x)) (\partial_{t} Φ (t, x)) = D h (Φ (t, x)) F (Φ (t, x))$ ; compositions of $C^{1}$ maps are $C^{1}$ ; differentiability conventions are those of total derivatives and Jacobians (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

Closed bounded real intervals are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line); continuous images of compact sets are compact, and open covers of compact sets have finite subcovers (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset). A differentiable real function whose derivative has a fixed nonzero sign on an interval is strictly monotone by The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ . Closure is the smallest closed superset in the indicated subspace (Interior, closure, boundary, exterior, derived set and isolated point in a topological space).

Proof

technique · direct; apply the implicit function theorem to the scalar function $(t,x)\mapsto h(\Phi(t,x))$ at both crossings of the section

1.1givenF1F4algebra

By [F1] there are an open interval $J_{0} ∋ t_{0}$ and an open set $A_{0} \subseteq W$ with $x_{0} \in A_{0}$ with $J_{0} \times A_{0} \subseteq Ω$ and $Φ (J_{0} \times A_{0}) \subseteq W$ (shrink $A_{0}$ if necessary, using that $Φ (t_{0}, x_{0}) = x_{1} \in W$ and continuity of $Φ$ ). Define $G : J_{0} \times A_{0} \to R$ , $G (t, x) : = h (Φ (t, x))$ ; it is $C^{1}$ by [F1] and [F4], and $G (t_{0}, x_{0}) = h (x_{1}) = 0, \partial_{t} G (t_{0}, x_{0}) = D h (x_{1}) F (x_{1}) \neq 0.$

2.1step 1.1F1F3

Apply [F3] to $G$ at $(t_{0}, x_{0})$ : there are open $J_{1} \subseteq J_{0}$ containing $t_{0}$ , open $A_{1} \subseteq A_{0}$ containing $x_{0}$ and a unique $C^{1}$ function $τ : A_{1} \to J_{1}$ with $τ (x_{0}) = t_{0}$ such that for $(t, x) \in J_{1} \times A_{1}$ $h (Φ (t, x)) = 0 ⟺ t = τ (x) .$ In particular $h (Φ (τ (x), x)) = 0$ for $x \in A_{1}$ , which is the first part of clause 1.

2.2step 1.12.1F1F2F3

Reverse-time construction. The field $- F$ is $C^{1}$ on $U$ , and $Ψ (t, y) : = Φ (- t, y)$ is its local flow on the reflected domain, because $s \mapsto Φ (- s, y)$ solves $y^{'} = - F (y)$ and every solution is a restriction of the corresponding maximal solution [F2]. Apply steps 1.1-2.1 to $- F$ at the initial point $x_{1}$ and time $t_{0}$ : the reversed reference solution satisfies $Ψ (t_{0}, x_{1}) = Φ (- t_{0}, x_{1}) = x_{0}$ by [F2], and its crossing derivative is $\partial_{t} (h (Ψ (t, y))) ∣_{(t_{0}, x_{1})} = D h (x_{0}) (- F (x_{0})) \neq 0$ . Hence there are an open interval $J_{2} ∋ t_{0}$ , an open set $B_{0} \subseteq W$ with $x_{1} \in B_{0}$ and a $C^{1}$ function $σ : B_{0} \to J_{2}$ with $σ (x_{1}) = t_{0}$ such that $h (Φ (- t, y)) = 0, (t, y) \in J_{2} \times B_{0} ⟺ t = σ (y) .$ Define $Q : B_{0} \to Σ$ , $Q (y) : = Φ (- σ (y), y)$ ; then $Q$ is $C^{1}$ (composition of the $C^{1}$ map $(t, y) \mapsto Φ (t, y)$ with $y \mapsto (- σ (y), y)$ , whose values lie in $Ω$ because $σ (y) \in J_{2}$ ) and $h (Q (y)) = 0$ for every $y \in B_{0}$ .

3.1step 2.1F1F4

Define $P (x) : = Φ (τ (x), x)$ for $x \in A_{1}$ . Then $P$ is $C^{1}$ as a composition of $C^{1}$ maps, and $P (x) \in Σ$ for every $x \in A_{1}$ by step 2.1. Since $P (x_{0}) = x_{1}$ , continuity of $P$ gives an open set $A_{2} \subseteq A_{1}$ with $x_{0} \in A_{2}$ and $P (A_{2}) \subseteq W$ .

3.2step 2.13.12.2F1F2F4

Shrinking the two neighbourhoods. Choose an open set $A \subseteq A_{2}$ with $x_{0} \in A$ , $τ (A) \subseteq J_{2}$ and $P (A) \subseteq B_{0}$ ; this is possible because $τ (x_{0}) = t_{0} \in J_{2}$ , $P (x_{0}) = x_{1} \in B_{0}$ and $τ, P$ are continuous. Define $B : = B_{0} \cap σ^{- 1} (J_{1}) \cap Q^{- 1} (A) .$ Then $B$ is open (preimages of open sets under the continuous maps $σ, Q$ ) and contains $x_{1}$ : $σ (x_{1}) = t_{0} \in J_{1}$ and $Q (x_{1}) = Φ (- t_{0}, x_{1}) = x_{0} \in A$ by [F2]. By construction $Q (B) \subseteq A$ and $σ (B) \subseteq J_{1}$ , so restricting $τ$ to $A$ and $σ$ to $B$ gives the $C^{1}$ functions of clauses 1 and 2, with $P : A \to Σ$ and $Q : B \to Σ$ both $C^{1}$ and mapping into $Σ$ (steps 2.1 and 2.2).

3.3step 2.13.12.23.2F2F3

First inverse identity. Let $x \in A \cap Σ$ and put $y : = P (x) \in B_{0}$ and $s : = τ (x) \in J_{2}$ . Since $h (x) = 0$ and $Φ (- s, y) = Φ (- τ (x), Φ (τ (x), x)) = x$ by [F2], we have $h (Φ (- s, y)) = 0$ ; the equivalence of step 2.2 with the pair $(s, y) \in J_{2} \times B_{0}$ therefore gives $σ (y) = s = τ (x)$ . Hence $σ (y) = τ (x) \in J_{1}$ , so $y \in σ^{- 1} (J_{1})$ , and $Q (y) = Φ (- σ (y), y) = Φ (- τ (x), y) = x \in A$ , so $y \in Q^{- 1} (A)$ ; therefore $y \in B$ . Thus $P (A \cap Σ) \subseteq B$ , and $Q (P (x)) = Φ (- σ (y), y) = x$ for every $x \in A \cap Σ$ .

3.4step 2.12.23.2F2F3

Second inverse identity. Let $y \in B \cap Σ \cap Q^{- 1} (A)$ and put $x : = Q (y) \in A \cap Σ$ (the value lies in $Σ$ by step 2.2). By [F2], $Φ (σ (y), x) = Φ (σ (y), Φ (- σ (y), y)) = y$ , and $h (y) = 0$ because $y \in Σ$ ; the equivalence of step 2.1 with the pair $(σ (y), x) \in J_{1} \times A_{1}$ therefore gives $σ (y) = τ (x)$ . Hence $P (Q (y)) = Φ (τ (x), x) = Φ (σ (y), x) = y$ .

3.5givenF1F4F5step 2.1choose

Assume the additional exclusion in clause 4. Choose $ε > 0$ with $3 ε < t_{0}$ sufficiently small that the reference paths for $0 \leq s \leq ε$ and $t_{0} - 2 ε \leq s \leq t_{0} + 2 ε$ lie in $W$ , and $D h (Φ (s, x_{0})) F (Φ (s, x_{0}))$ has a fixed nonzero sign on each of these two intervals. Continuity and compactness give an open neighbourhood $A_{3}$ of $x_{0}$ such that all these paths for $x \in A_{3}$ remain in $W$ with the same respective signs. The compact middle path $K = {Φ (s, x_{0}) : ε \leq s \leq t_{0} - ε}$ lies in the open set $U ∖ {\overline{Σ}}^{U}$ by the additional hypothesis. By openness of $Ω$ , continuity of $Φ$ and finite product-neighbourhood covers of the compact time intervals, further shrink $A_{3}$ so that $[0, t_{0} + 2 ε] \times A_{3} \subseteq Ω$ and $Φ ([ε, t_{0} - ε] \times A_{3}) \subseteq U ∖ {\overline{Σ}}^{U}$ . Also shrink it so that $∣ τ (x) - t_{0} ∣ < ε$ .

4.1givenF3F4F5step 3.2

Since $D h (x_{i}) F (x_{i}) \neq 0$ , one has $D h (x_{i}) \neq 0$ at each $x_{i}$ ( $i = 0, 1$ ). For $n \geq 2$ , after permuting coordinates, [F3] makes $Σ$ near $x_{i}$ the graph $x_{n} = f_{i} (x_{1}, \dots, x_{n - 1})$ of a $C^{1}$ function; its graph parametrization and coordinate projection are an embedded section chart. For $n = 1$ , continuity of $h^{'}$ and $h^{'} (x_{i}) \neq 0$ give an interval on which $h^{'}$ has a fixed nonzero sign; the mean value theorem [F5] makes $h$ strictly monotone there, so $Σ$ in that interval is the singleton ${x_{i}}$ , a zero-dimensional section chart. Shrink $A$ into the chart near $x_{0}$ and shrink $B_{0}$ into the target chart near $x_{1}$ ; by continuity of $P$ shrink $A$ further so $P (A) \subseteq B_{0}$ , then repeat the construction of $B$ in step 3.2. In dimensions $n \geq 2$ , the chart coordinates of $P : A \to Σ$ and $Q : B \to Σ$ are ambient coordinate projections composed with ambient $C^{1}$ maps, so both are $C^{1}$ ; for $n = 1$ these maps are locally constant into the singleton charts. Their restrictions to the sections are therefore $C^{1}$ in the same charts. The inverse identities of steps 3.3--3.4 remain valid after these neighbourhood restrictions.

5.1step 3.13.23.33.44.1F1F4

By steps 3.3 and 3.4, $P (A \cap Σ) = B \cap Σ \cap Q^{- 1} (A)$ : if $y = P (x)$ with $x \in A \cap Σ$ , then $y \in B$ , $y \in Σ$ and $Q (y) = x \in A$ ; conversely, if $y \in B \cap Σ \cap Q^{- 1} (A)$ , then $x : = Q (y) \in A \cap Σ$ and step 3.4 gives $P (x) = y$ . Since $B$ and $Q^{- 1} (A)$ are open in $R^{n}$ , the set $B \cap Σ \cap Q^{- 1} (A)$ is open in the relative topology of $Σ$ . On $P (A \cap Σ)$ the maps $P$ and $Q$ are inverse to each other by steps 3.3 and 3.4, and both are $C^{1}$ in the charts of step 4.1, so $P : A \cap Σ \to P (A \cap Σ)$ is a $C^{1}$ diffeomorphism with inverse $Q ∣_{P (A \cap Σ)}$ . This proves clauses 2 and 3.

6.1step 3.53.23.33.45.1F4F5∎

Repeat step 3.2 with $A \subseteq A_{3}$ , retaining its inverse identities and charts. For $x \in A \cap Σ$ , the scalar function $s \mapsto h (Φ (s, x))$ is strictly monotone on $[0, ε]$ and vanishes at $0$ , hence has no zero for $0 < s \leq ε$ . The middle interval has no $Σ$ -hits by step 3.5. On $[t_{0} - 2 ε, t_{0} + 2 ε]$ the scalar function is strictly monotone and has its unique zero at $τ (x)$ ; thus it has no zero on $[t_{0} - ε, τ (x))$ . These intervals cover $(0, τ (x))$ , and $τ (x) > t_{0} - ε > 0$ . This proves the first-return conclusion of clause 4, together with clauses 1-3 already proved.

Remarks

The two transversality hypotheses are exactly what the two applications of the implicit function theorem use: $D h (x_{1}) F (x_{1}) \neq 0$ at the forward crossing and $D h (x_{0}) F (x_{0}) \neq 0$ at the crossing of the reversed reference orbit. If only the forward crossing were transversal, the return time $τ$ would still exist as a $C^{1}$ function, but the construction of the inverse and hence of the local conjugacy of $P$ with the reversed flow would not be available.
No periodicity of the reference orbit is assumed: $x_{1}$ need not equal $x_{0}$ . When $x_{0}$ is periodic of period $t_{0}$ and the section is transversal at $x_{0}$ , the map $P$ above is a local return map; it is a first-return map when the exclusion hypothesis of clause 4 holds, and every fixed point of $P$ corresponds to a periodic orbit of the flow through the section in the usual way; that correspondence is not needed in this batch.

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Finite-time tangent growth rates and Lyapunov exponents of a C¹ flow

Definition

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, let $F : U \to R^{n}$ be of class $C^{1}$ ( $C^{k}$ Euclidean maps and diffeomorphisms), and let $Φ : Ω \to U$ be the local flow of $x^{'} = F (x)$ (The local flow of an autonomous first-order system); recall that $Ω$ is open and $Φ$ is $C^{1}$ on $Ω$ (The flow domain of a C¹ autonomous field is open and the flow is C¹ on it).

Let $x \in U$ and $t > 0$ with $(t, x) \in Ω$ , and let $u \in R^{n} ∖ {0}$ be such that $D_{x} Φ_{t} (x) u \neq 0$ , where $D_{x} Φ_{t} (x)$ is the derivative of the map $y \mapsto Φ (t, y)$ at $x$ (The local flow of an autonomous first-order system, $C^{k}$ Euclidean maps and diffeomorphisms). The finite-time tangent growth rate of $(x, u)$ over the time $t$ is

$r (t, x, u) : = \frac{1}{t} \log ⁣ (\frac{∥ D_{x} Φ_{t} (x) u ∥_{2}}{∥ u ∥_{2}}) \in R,$

the Euclidean norms being those of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\log$ the natural logarithm (The natural logarithm as the inverse of the exponential function). The expression is a real number because $∥ u ∥_{2} > 0$ and $∥ D_{x} Φ_{t} (x) u ∥_{2} > 0$ .

For the infinite-time notion require the forward lifetime $I_{x} = {t : (t, x) \in Ω}$ to be unbounded above. Since $I_{x}$ is an interval containing $0$ , this means that $(t, x) \in Ω$ for every $t \geq 0$ . Under this additional condition, if the limit

$λ (x, u) : = \lim_{\begin{array}{c} t \to \infty \\ (t, x) \in Ω \end{array}} r (t, x, u)$

exists in $R$ , it is called the Lyapunov exponent of $(x, u)$ . If the forward lifetime is bounded above or the limit does not exist, no Lyapunov exponent is assigned; in particular no value $+ \infty$ or $- \infty$ is named by this definition.

Remarks

The derivative $D_{x} Φ_{t} (x)$ is invertible whenever it is defined: on the open set $Ω$ , the maps $Φ_{t}$ and $Φ_{- t}$ are inverse to each other by The local flow is a partial one-parameter group: identity, composition and inverse-time laws, and the chain rule applied to $Φ_{- t} \circ Φ_{t} = i d$ gives $D Φ_{- t} (Φ_{t} (x)) \circ D_{x} Φ_{t} (x) = i d$ . Hence the admissibility condition $D_{x} Φ_{t} (x) u \neq 0$ holds automatically for $u \neq 0$ at every point where $D_{x} Φ_{t} (x)$ is defined; it is recorded so that the definition does not depend on the invertibility statement.
The rate is homogeneous in the direction: $r (t, x, c u) = r (t, x, u)$ for every real $c \neq 0$ , since both norms scale by $∣ c ∣$ . Thus opposite directions have the same rate, and the quantity may be negative, zero or positive, and may depend on both $x$ and the direction $u$ .
For a linear field $F (x) = A x$ the flow is $Φ (t, x) = e^{t A} x$ wherever it is defined, so $D_{x} Φ_{t} (x) = e^{t A}$ is independent of $x$ and the finite-time rate reduces to $t^{- 1} \log (∥ e^{t A} u ∥_{2} / ∥ u ∥_{2})$ . Whether the limit exists is not claimed by this definition; it is a property of $A$ and $u$ .

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The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets

Definition

Let

$X : = {0, 1}^{Z} = {s : s is a function Z \to {0, 1}}$

be the set of bi-infinite binary sequences, with the function-set convention of The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = {0, 1, \dots, n - 1}$ ; for $s \in X$ write $s_{k} : = s (k)$ for the value at the integer $k$ . Define

$d (s, t) : = \sum_{k = 0}^{\infty} 2^{- k} ∣ s_{k} - t_{k} ∣ + \sum_{k = 1}^{\infty} 2^{- k} ∣ s_{- k} - t_{- k} ∣ (s, t \in X) .$

Both series converge by comparison with the geometric series $\sum_{k \geq 0} 2^{- k} = 2$ (Series, partial sums, convergence and the sum, divergence, and the tail series, For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, Integer powers $a^{m}$ , Absolute value in an ordered field), since $∣ s_{j} - t_{j} ∣ \in {0, 1}$ for every $j \in Z$ ; hence $d (s, t)$ is a well-defined real number. That $d$ is a metric on $X$ is proved in the companion lemma The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact ↗ and is not part of this definition.

The (bilateral) shift is the map

$σ : X ⟶ X, σ (s)_{k} : = s_{k + 1} (k \in Z, s \in X) .$

For $n \geq 1$ the $n$ -fold iterate is written $σ^{n}$ , defined by $σ^{1} : = σ$ and $σ^{n + 1} : = σ \circ σ^{n}$ ; a point $s \in X$ is periodic when $σ^{n} (s) = s$ for some $n \geq 1$ .

A cylinder is a set of the form

$[w]_{p, q} : = {s \in X : s_{k} = w_{k} for p \leq k \leq q},$

where $p \leq q$ are integers and $w = (w_{p}, \dots, w_{q}) \in {0, 1}^{{p, \dots, q}}$ is a finite binary word; the integers $p, q$ and the word $w$ are part of the data. When $p = q$ this is the set of sequences with prescribed value at the single coordinate $p$ .

Remarks

The set $X$ is uncountable and the map $σ$ is a bijection with inverse $σ^{- 1} (s)_{k} = s_{k - 1}$ ; the definition itself asserts only the displayed formulas, while continuity of $σ$ is a theorem (The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense).
No measure, probability or invariant measure on $X$ is introduced by this definition. The example used in this collection is topological; statements about invariant measures for the one-sided fair-coin shift are separate published results and are not consumed here.
Cylinders form a basis for the metric topology. More precisely, for $m \geq 0$ the centered cylinder $[s]_{- m, m}$ satisfies $B (s, 2^{- m}) \subseteq [s]_{- m, m} \subseteq \overline{B} (s, 2^{1 - m})$ . An arbitrary cylinder $[w]_{p, q}$ contains $B (s, 2^{- \max {∣ p ∣, ∣ q ∣}})$ about each of its points $s$ , but need not have a comparable outer radius (The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact ↗, clause 2 and proof step 2.1).

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The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact

Statement

Let $X = {0, 1}^{Z}$ , let $d$ and $σ$ be the metric expression and the shift map of The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets, and for $m \geq 0$ let $F_{m} : = {- m, \dots, m}$ and $[s]_{F_{m}} : = {t \in X : t_{k} = s_{k} for k \in F_{m}}$ be the cylinder determined by $s$ on $F_{m}$ . Then:

$d$ is a metric on $X$ (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric);
for every $s \in X$ and every $m \geq 0$ ,

$B (s, 2^{- m}) \subseteq [s]_{F_{m}} \subseteq \overline{B} (s, 2^{1 - m});$

consequently every cylinder is open in the metric topology (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement) and every nonempty open subset of $X$ contains a cylinder; 3. $X$ is sequentially compact (Countably compact, sequentially compact and limit point compact metric spaces): every sequence in $X$ has a subsequence converging in $d$ (Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ), and the construction of that subsequence uses no choice principle; 4. granting the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and the Axiom of Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain), $(X, d)$ is compact (Open cover, subcover, compact metric space, and compact subset of a metric space).

Facts & Assumptions

Given: The set $X$ of bi-infinite binary sequences, the expression $d (s, t) = \sum_{k \geq 0} 2^{- k} ∣ s_{k} - t_{k} ∣ + \sum_{k \geq 1} 2^{- k} ∣ s_{- k} - t_{- k} ∣$ , and the cylinders $[s]_{F_{m}}$ .

[F1]

The real absolute value satisfies $∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$ (The triangle inequality). The geometric series satisfies $\sum_{k \geq 0} 2^{- k} = 2$ and $\sum_{k = m + 1}^{\infty} 2^{- k} = 2^{- m}$ for $m \geq 0$ ; a series of nonnegative reals converges if and only if its partial sums are bounded, and convergent series may be added termwise and compared termwise (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum, Convergent series add and scale termwise, Series, partial sums, convergence and the sum, divergence, and the tail series, Lower bound, bounded below, bounded set).

[F2]

Balls and topology: $B (s, ε) = {t : d (s, t) < ε}$ and $\overline{B} (s, ε) = {t : d (s, t) \leq ε}$ for the metric $d$ ; a set is open when every point of it has a ball contained in it; a metric space carries the topology induced by its balls (Open ball, closed ball and sphere in a metric space, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

[F3]

A subset $D \subseteq X$ is dense when every nonempty open subset meets $D$ (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets).

[F4]

A metric space is sequentially compact when every sequence has a convergent subsequence; compact, countably compact, limit point compact and sequentially compact metric spaces coincide, and this equivalence is proved under the Axioms of Countable and Dependent Choice (Open cover, subcover, compact metric space, and compact subset of a metric space, Countably compact, sequentially compact and limit point compact metric spaces, For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice, Sequences of reals: bounded, eventually, frequently, tails, subsequences).

Proof

technique · direct; verify the metric axioms, identify cylinders with balls up to a factor two, and diagonalise a sequence of binary sequences

1.1givenF1F2algebra

The expression $d (s, t)$ is a sum of two convergent series of nonnegative terms by [F1], since $∣ s_{j} - t_{j} ∣ \leq 1$ ; hence it is a nonnegative real and $d (s, t) = d (t, s)$ because $∣ s_{j} - t_{j} ∣ = ∣ t_{j} - s_{j} ∣$ . If $s \neq t$ there is $j \in Z$ with $s_{j} \neq t_{j}$ ; the corresponding term of $d (s, t)$ is positive and all terms are nonnegative, so $d (s, t) > 0$ . Finally, for $s, t, u \in X$ the termwise inequality $∣ s_{j} - t_{j} ∣ \leq ∣ s_{j} - u_{j} ∣ + ∣ u_{j} - t_{j} ∣$ and [F1] give $d (s, t) \leq d (s, u) + d (u, t)$ after multiplying by $2^{- ∣ j ∣}$ and adding the two one-sided series. Hence $d$ is a metric, which is clause 1.

1.2givenF1F2algebra

Fix $s \in X$ and $m \geq 0$ . If $t \in B (s, 2^{- m})$ then $2^{- ∣ j ∣} \leq d (s, t) < 2^{- m}$ for every $j$ with $∣ j ∣ \leq m$ and $t_{j} \neq s_{j}$ ; since $2^{- ∣ j ∣} \geq 2^{- m}$ there, no such $j$ exists, so $t \in [s]_{F_{m}}$ . Conversely, if $t \in [s]_{F_{m}}$ then all terms with $∣ j ∣ \leq m$ vanish and the remaining terms are at most $2^{- ∣ j ∣}$ ; hence $d (s, t) \leq \sum_{∣ j ∣ > m} 2^{- ∣ j ∣} = 2 \cdot 2^{- m} = 2^{1 - m}$ , that is $t \in \overline{B} (s, 2^{1 - m})$ . This proves the two inclusions of clause 2.

1.3givenF1F2

Let $(x_{n})_{n \geq 0}$ be a sequence in $X$ . Define $F_{m} = {- m, \dots, m}$ and construct subsequences $(x_{n}^{(m)})_{n \geq 0}$ recursively: $x_{n}^{(0)} : = x_{n}$ , and given $(x_{n}^{(m - 1)})$ , the map $n \mapsto x_{n}^{(m - 1)} ∣_{F_{m}}$ takes values in the finite set ${0, 1}^{F_{m}}$ , so some value $r_{m}$ occurs for infinitely many $n$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences); let $r_{m}$ be the lexicographically least such value (with respect to a fixed enumeration of $F_{m}$ ), and let $i_{0}^{(m)} < i_{1}^{(m)} < \dots$ enumerate the infinite set ${n : x_{n}^{(m - 1)} ∣_{F_{m}} = r_{m}}$ ; define $x_{j}^{(m)} : = x_{i_{j}^{(m)}}^{(m - 1)}$ . This is a simultaneous recursive construction with no choice: each $r_{m}$ is selected by a least-element rule, and each increasing enumeration is defined by least-element recursion on an infinite subset of $N$ . For $m \geq 1$ every term of $(x_{n}^{(m)})$ has restriction $r_{m}$ to $F_{m}$ , and each $(x_{n}^{(j)})$ with $j \geq m$ is a termwise subsequence of $(x_{n}^{(m)})$ .

2.1step 1.2F2F3algebra

Every cylinder $[w]_{p, q}$ is open: if $t \in [w]_{p, q}$ and $M : = \max {∣ p ∣, ∣ q ∣}$ , then any $u$ with $d (t, u) < 2^{- M}$ satisfies $u_{j} = t_{j}$ for every $∣ j ∣ \leq M$ by the argument of step 1.2, hence $u \in [w]_{p, q}$ ; thus $B (t, 2^{- M}) \subseteq [w]_{p, q}$ . Conversely, if $V \subseteq X$ is nonempty and open and $s \in V$ , choose $ε > 0$ with $B (s, ε) \subseteq V$ and $m$ with $2^{1 - m} < ε$ ; then step 1.2 gives $[s]_{F_{m}} \subseteq \overline{B} (s, 2^{1 - m}) \subseteq B (s, ε) \subseteq V$ . Hence cylinders are open and form a basis for the topology, as claimed in clause 2.

2.2step 1.21.3F1F2algebra

Put $y_{j} : = x_{j}^{(j)}$ ; the indices of the original sequence at which $y_{j}$ sits are strictly increasing in $j$ , so $(y_{j})$ is a subsequence. For $m \geq 1$ , if $j, j^{'} \geq m$ , then $y_{j}$ and $y_{j^{'}}$ both have restriction $r_{m}$ to $F_{m}$ by step 1.3, so $d (y_{j}, y_{j^{'}}) \leq \sum_{∣ k ∣ > m} 2^{- ∣ k ∣} = 2^{1 - m}$ ; hence $(y_{j})$ is a Cauchy sequence in $(X, d)$ . Define $s \in X$ by $s_{k} : = r_{m} (k)$ for every $m \geq \max {1, ∣ k ∣}$ ; this is well defined because $r_{m + 1} ∣_{F_{m}} = r_{m}$ for $m \geq 1$ (every term of $(x_{n}^{(m + 1)})$ is a term of $(x_{n}^{(m)})$ , so the restriction of $r_{m + 1}$ to $F_{m}$ stabilises the same value). Then $d (y_{j}, s) \leq 2^{1 - m}$ for every $j \geq m \geq 1$ , so $y_{j} \to s$ by [F2]. This proves clause 3.

3.1step 2.2F4∎

Clause 4 follows from clause 3 and [F4], which turns sequential compactness into compactness under the two stated choice axioms.

Remarks

The metric can be written with the symmetric weight $2^{- ∣ k ∣}$ : for $s, t \in X$ , splitting the sum at $k = 0$ shows $d (s, t) = \sum_{k \in Z} 2^{- ∣ k ∣} ∣ s_{k} - t_{k} ∣$ in the sense that the two defining one-sided series are exactly the positive- and negative-index parts. The one-sided form is used here because it avoids a two-sided summation convention.
The diagonal construction of step 1.3 is canonical: the value $r_{m}$ is the least stabilised restriction, and the increasing enumeration of an infinite subset of $N$ is produced by least-element recursion. Countable choice is used only when compactness is read off from sequential compactness in clause 4, which is why that clause is stated as a separate hypothesis.

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Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list

Definition

Let $(X, d)$ be a metric space (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric) with its metric topology (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement) and let $f : X \to X$ be a map. For $n \geq 1$ write $f^{n}$ for the $n$ -fold iterate, $f^{1} : = f$ and $f^{n + 1} : = f \circ f^{n}$ .

$f$ is topologically mixing when for all nonempty open sets $U, V \subseteq X$ there is an integer $N$ such that

$f^{n} (U) \cap V \neq \emptyset for every n \geq N .$

A point $x \in X$ is a periodic point of $f$ when $f^{n} (x) = x$ for some $n \geq 1$ . The set of periodic points of $f$ is written $Per (f)$ . The periodic points are dense in $X$ when $Per (f)$ is dense in $X$ , that is (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets) when every nonempty open subset of $X$ meets $Per (f)$ .
$f$ has sensitive dependence on initial conditions when there is a real number $δ > 0$ such that for every $x \in X$ and every $ε > 0$ there are $y \in X$ and $n \geq 1$ with

$d (x, y) < ε and d (f^{n} (x), f^{n} (y)) \geq δ .$

Such a number $δ$ is called a sensitivity constant for $f$ .

Finally, say that $f$ is chaotic in the stated sense when $f$ is continuous (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form), topologically mixing, has dense periodic points, and has sensitive dependence on initial conditions.

Remarks

The list is a definition of a chosen package of properties, not a definition of the word "chaos" in all of its uses. Devaney's closely related list replaces topological mixing by topological transitivity; mixing implies transitivity, while this definition makes no converse claim. Positive topological entropy, Li-Yorke chaos, and measure-theoretic mixing are further distinct properties that this definition neither implies nor assumes.
Topological mixing immediately implies the open-set form of topological transitivity: for nonempty open $U, V$ the definition supplies $N$ with $f^{N} (U) \cap V \neq \emptyset$ . Sensitivity is required with one constant $δ$ uniform over all points and all radii $ε$ ; it is a statement about the family of iterates, not about a single pair of orbits.
The definition applies to non-invertible maps; only forward iterates are used. When $f$ is a bijection, applying the same definitions to $f^{- 1}$ gives a separate statement, and nothing here asserts two-sided mixing.

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The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense

Statement

Let $X = {0, 1}^{Z}$ , let $d$ be the metric of The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact and let $σ : X \to X$ , $σ (s)_{k} = s_{k + 1}$ , be the bilateral shift of The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets. Then:

$σ$ is continuous (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form);
$σ$ is topologically mixing;
the periodic points of $σ$ are dense in $X$ ;
$σ$ has sensitive dependence on initial conditions with sensitivity constant $δ = 1$ ;

consequently $σ$ is chaotic in the stated sense (Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list).

Facts & Assumptions

Given: The shift space $(X, d)$ , the shift map $σ$ and the cylinders $[w]_{p, q} = {s : s_{k} = w_{k} for p \leq k \leq q}$ .

[F1]

$d$ is a metric; every cylinder is open, and every nonempty open subset of $X$ contains a cylinder (The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

[F2]

$σ$ is a bijection with $σ (s)_{k} = s_{k + 1}$ and inverse $σ^{- 1} (s)_{k} = s_{k - 1}$ ; for $m \geq 1$ , $σ^{m} (s)_{k} = s_{k + m}$ (The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets).

[F3]

For $s, t \in X$ , the defining series of $d$ are nonnegative and summable with $\sum_{k \geq 0} 2^{- k} = 2$ and $\sum_{k \geq m + 1} 2^{- k} = 2^{- m}$ (The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets, For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum, Series, partial sums, convergence and the sum, divergence, and the tail series).

[F4]

A map $g : X \to X$ is continuous at $s$ when for every $ε > 0$ there is $δ > 0$ with $g (B (s, δ)) \subseteq B (g (s), ε)$ (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, Open ball, closed ball and sphere in a metric space).

[F5]

A subset $D \subseteq X$ is dense when every nonempty open subset meets $D$ (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets).

[F6]

The list of properties defining "chaotic in the stated sense" is in Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list: continuity, topological mixing, dense periodic points and sensitive dependence.

Proof

technique · direct; use cylinders as a basis and prescribe finitely many coordinates

1.1givenF1F2F3F4algebra

For $s, t \in X$ and any $m \geq 0$ , if $s_{j} = t_{j}$ for $∣ j ∣ \leq m + 1$ , then $σ (s)_{j} = σ (t)_{j}$ for $∣ j ∣ \leq m$ ; hence $σ ([s]_{F_{m + 1}}) \subseteq [σ (s)]_{F_{m}}$ . By clause 2 of [F1], $B (s, 2^{- (m + 1)}) \subseteq [s]_{F_{m + 1}}$ and $[σ (s)]_{F_{m}} \subseteq \overline{B} (σ (s), 2^{1 - m})$ , so $d (s, t) < 2^{- (m + 1)}$ implies $d (σ s, σ t) \leq 2^{1 - m}$ ; given $ε > 0$ choose $m$ with $2^{1 - m} < ε$ and $δ = 2^{- (m + 1)}$ . This proves continuity at every $s$ , which is clause 1.

1.2givenF1F2algebra

Let $U, V \subseteq X$ be nonempty and open. By [F1] there are cylinders $[w]_{p, q} \subseteq U$ and $[w^{'}]_{p^{'}, q^{'}} \subseteq V$ . Let $n$ be an integer with $(p, \dots, q) \cap (p^{'} + n, \dots, q^{'} + n) = \emptyset$ ; this holds for every $n > q - p^{'}$ . Define $s \in X$ by $s_{k} : = w_{k}$ for $p \leq k \leq q$ , $s_{k} : = w_{k - n}^{'}$ for $p^{'} + n \leq k \leq q^{'} + n$ , and $s_{k} : = 0$ for all remaining $k$ ; the two prescriptions are compatible because their index sets are disjoint. Then $s \in [w]_{p, q}$ , and for $j \in [p^{'}, q^{'}]$ one has $σ^{n} (s)_{j} = s_{j + n} = w_{j}^{'}$ , so $σ^{n} (s) \in [w^{'}]_{p^{'}, q^{'}}$ . Hence $σ^{n} (U) \cap V \neq \emptyset$ for every $n > q - p^{'}$ , which exhibits the integer $N$ required by topological mixing.

1.3givenF1F2algebra

Let $U \subseteq X$ be nonempty and open and choose a cylinder $[w]_{p, q} \subseteq U$ by [F1]; put $L : = q - p + 1 \geq 1$ and define $s \in X$ by $s_{k} : = w_{p + ((k - p) m o d L)}$ , the $L$ -periodic extension of the word $w$ . Then $s_{k} = w_{k}$ for $p \leq k \leq q$ , so $s \in [w]_{p, q} \subseteq U$ , and $s_{k + L} = s_{k}$ for every $k$ ; by [F2], $σ^{L} (s)_{k} = s_{k + L} = s_{k}$ for all $k$ , so $s$ is a periodic point of $σ$ . Thus every nonempty open set meets $Per (σ)$ , and [F5] gives clause 3.

1.4givenF2F3algebra

Let $x \in X$ and $ε > 0$ ; choose an integer $k \geq 1$ with $2^{- k} < ε$ and define $y \in X$ by $y_{j} : = x_{j}$ for $j \neq k$ and $y_{k} : = 1 - x_{k}$ . Then $d (x, y) = 2^{- k} < ε$ because exactly one term of the defining sum is nonzero and equal to $2^{- k}$ . Moreover $σ^{k} (x)_{0} = x_{k} \neq y_{k} = σ^{k} (y)_{0}$ , so in the defining sum for $d (σ^{k} x, σ^{k} y)$ the term with index $0$ equals $2^{0} \cdot 1 = 1$ and all terms are nonnegative; hence $d (σ^{k} x, σ^{k} y) \geq 1$ . Therefore $δ = 1$ is a sensitivity constant, which is clause 4.

2.1step 1.11.21.31.4F6∎

Clauses 1-4 and [F6] give that $σ$ is continuous, topologically mixing, has dense periodic points and is sensitive; by definition, $σ$ is chaotic in the stated sense.

Remarks

Compactness of $(X, d)$ is not used in the proof: mixing, density of periodic points and sensitivity are checked directly from cylinders. The companion lemma The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact proves that $X$ is sequentially compact, and compact under the Axioms of Countable and Dependent Choice, so the example also matches the usual convention in which a chaotic system is a continuous self-map of a compact metric space.
The integer $k$ in step 1.4 is positive, so the witness sequence $y$ is obtained by flipping a coordinate in the future tail; flipping a coordinate at index $k \leq 0$ would instead give $d (x, y) = 2^{- ∣ k ∣}$ but the displayed iterate argument uses $σ^{k}$ with $k \geq 1$ . The sensitivity constant $1$ is not claimed to be optimal: any $0 < δ \leq 1$ is also a sensitivity constant, and no statement is made about the largest such constant.

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Scope of Poincaré recurrence for mechanical systems: finite invariant volume and complete return maps

Statement

The recurrence content attached to this page is supplied by published items and is subject to their stated hypotheses.

In a measure-preserving system $(X, A, μ, T)$ with $μ (X) < \infty$ , for every $E \in A$ almost every point of $E$ returns to $E$ under $T^{n}$ for infinitely many positive integers $n$ (Poincare recurrence for finite measure-preserving systems, Measure-preserving transformations and systems). If $X$ is second countable with a countable basis in $A$ , then outside one null set every neighbourhood of every point is revisited infinitely often, and in the metrizable case there is a strictly increasing sequence $n_{j}$ with $T^{n_{j}} x \to x$ (Topological recurrence on second-countable spaces). For a Hamiltonian flow, a measurable invariant region of finite symplectic volume on which a nonzero time map and all of its iterates are defined satisfies the same recurrence conclusion for that time map (Poincaré recurrence for finite-volume Hamiltonian invariant regions).
The following qualifications are part of the statement and are not removable.

Finiteness of measure is a hypothesis. The conclusion fails in general for infinite measure: for $T (x) = x + 1$ on $R$ with Lebesgue measure, the set $E = (0, 1 / 2)$ has positive measure and no point of $E$ returns to $E$ at any positive integer time, so the analogue of clause 1 is false without $μ (X) < \infty$ . For a concrete mechanical contrast take the free-particle Hamiltonian $H (q, p) = p^{2} / (2 m)$ on $X = R_{q} \times R_{p}$ with $m > 0$ , phase volume $d q d p$ , and complete flow $ϕ_{t} (q, p) = (q + t p / m, p)$ . Its time-one map preserves this infinite volume (each fixed- $p$ slice is a translation in $q$ ). In $E_{f r e e} = (0, 1 / 2) \times (m, 2 m)$ every positive integer iterate has $q_{n} = q + n p / m > 1$ , so no point returns to this positive-volume set. Thus complete, volume-preserving mechanical motion on the full unconfined phase space need not be recurrent; the finite-invariant-volume hypothesis is essential.
The return map must be defined for all iterates. The time- $τ$ map of a Hamiltonian flow is a measure-preserving map only where it is defined: the invariant region must be contained in the domain of $ϕ_{τ}$ and of every iterate $ϕ_{n τ}$ . For a local flow in the sense of The local flow of an autonomous first-order system this is an additional hypothesis on the region, not a consequence of the flow being locally defined; on a compact energy level of a complete bounded system it must be verified from the model.
Recurrence is an almost-everywhere statement about a measure, not about every trajectory. The null exceptional set depends on the measurable set being revisited; a specified trajectory, or a finite-time observation, is not claimed to return. The topological corollary supplies a single null set only after a countable basis is fixed and one grants its countable-choice hypotheses.
No identification with the shift example. The bilateral shift of The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense is a topological example on the symbol space; no claim is made here that it arises as the return map of a smooth mechanical system, and no claim about any physical system is made by this remark.

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Exponential dichotomy for a hyperbolic endomorphism: invariant stable and unstable subspaces with exponential estimates

Statement

Let $V$ be a real vector space of finite dimension $n \geq 1$ , equipped with a symmetric positive-definite bilinear form $⟨ \cdot, \cdot ⟩ : V \times V \to R$ , and set $∥ v ∥_{2} : = \sqrt{⟨ v, v ⟩}$ . Let $T : V \to V$ be linear, let $V_{C} : = C \otimes_{R} V$ with canonical embedding $ι v : = 1 \otimes v$ and complex-linear extension $T_{C} : = {id}_{C} \otimes T$ (Complexification as $C \otimes_{R} V$ with its canonical real-linear embedding, Complexification of a real-linear map), and use the exponential series $e^{t T}$ and $e^{t T_{C}}$ of these bounded operators (The exponential series of a bounded operator; boundedness is justified in [F1]). Suppose that no eigenvalue of $T_{C}$ in the sense of Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism has real part $0$ .

Then there exist $T$ -invariant linear subspaces $E_{s}, E_{u} \subseteq V$ and real constants $C \geq 1$ , $c > 0$ such that $V = E_{s} \oplus E_{u}$ and, for every $t \in R$ :

$∥ e^{t T} v ∥_{2} \leq C e^{- c t} ∥ v ∥_{2}$ for $v \in E_{s}$ and $t \geq 0$ ;
$∥ e^{t T} v ∥_{2} \leq C e^{c t} ∥ v ∥_{2}$ for $v \in E_{u}$ and $t \leq 0$ ;
$∥ e^{t T} v ∥_{2} \geq C^{- 1} e^{c t} ∥ v ∥_{2}$ for $v \in E_{u}$ and $t \geq 0$ ;
$E_{s} = {v \in V : e^{t T} v \to 0 (t \to + \infty)}$ and $E_{u} = {v \in V : e^{t T} v \to 0 (t \to - \infty)}$ .

Facts & Assumptions

Given: The finite-dimensional real vector space $V$ with its symmetric positive-definite bilinear form, the linear map $T$ , its complexification $T_{C}$ , and the assumption that no complex eigenvalue of $T_{C}$ is purely imaginary.

[F1]

The real spectral theorem applied to ${id}_{V}$ supplies a real orthonormal basis of $V$ (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis). In that basis every element of $V_{C}$ has a unique form $x + i y : = ι x + i ι y$ , with $x, y \in V$ . Define $⟨ x + i y, x^{'} + i y^{'} ⟩_{C} : = ⟨ x, x^{'} ⟩ + ⟨ y, y^{'} ⟩ + i (⟨ y, x^{'} ⟩ - ⟨ x, y^{'} ⟩)$ and $∥ x + i y ∥_{2, C} : = (∥ x ∥_{2}^{2} + ∥ y ∥_{2}^{2})^{1 / 2}$ ; this is the induced Hermitian inner product and norm, and $ι$ is isometric. These coordinates identify $V$ with $R^{n}$ and $V_{C}$ with $R^{2 n}$ , so both spaces are complete ( $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ , $C$ is the real coordinate plane, with coordinate arithmetic). If a linear map has coordinate matrix $L = (L_{i j})$ , Cauchy--Schwarz row by row gives $∥ L z ∥_{2}^{2} \leq (\sum_{i, j} ∣ L_{i j} ∣^{2}) ∥ z ∥_{2}^{2}$ in the corresponding real coordinates (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation); hence $T$ and $T_{C}$ are bounded. Thus the exponential-series lemma applies to both; it gives absolute operator-norm convergence, $E (0) = id$ , $E (s + t) = E (s) E (t)$ , $E (t)^{- 1} = E (- t)$ and $E^{'} (t) = T_{C} E (t) = E (t) T_{C}$ . Termwise application of $T_{C}^{m} ι = ι T^{m}$ gives $E (t) ι = ι e^{t T}$ (The exponential series of a bounded operator, Complexification as $C \otimes_{R} V$ with its canonical real-linear embedding, Complexification of a real-linear map).

[F2]

The minimal polynomial $μ_{T} \in R [x]$ annihilates $T$ , the characteristic polynomial satisfies $χ_{T_{C}} = χ_{T}$ , and $μ_{T}$ and $χ_{T}$ have the same irreducible factors over $C$ ; every nonconstant complex polynomial splits into linear factors, so the eigenvalues of $T_{C}$ are exactly the complex roots of $μ_{T}$ , each with a positive multiplicity $e_{λ}$ equal to its multiplicity as a root of $μ_{T}$ (The annihilator set $Ann (T) = {p \in F [x] : p (T) = 0}$ ; once existence is proved, its unique monic generator $μ_{T}$ is the minimal polynomial, Complexification preserves the characteristic and minimal polynomials of a finite-dimensional real operator, The minimal and characteristic polynomials have exactly the same monic irreducible factors, Fundamental theorem of algebra: every nonconstant complex polynomial has a complex root, Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism).

[F3]

Since $μ_{T_{C}} = μ_{T}$ splits over $C$ , the generalised eigenspaces $G_{λ} = \ker (T_{C} - λ)^{e_{λ}}$ over the distinct eigenvalues $λ$ of $T_{C}$ form a direct sum decomposition $V_{C} = ⨁_{λ} G_{λ}$ , each $G_{λ}$ being $T_{C}$ -invariant (If the minimal polynomial splits, $V$ is the direct sum of the stabilised generalised eigenspaces). On each $G_{λ}$ the operator admits a Jordan basis, so $T_{C} ∣_{G_{λ}} = λ I + N_{λ}$ with $N_{λ}$ nilpotent of index at most $e_{λ}$ (Jordan canonical form from the elementary divisors of $V_{T}$ ).

[F4]

Coprime polynomials satisfy a Bézout identity: if $p, q \in F [x]$ have no common root, there are $a, b \in F [x]$ with $a p + b q = 1$ (Bézout identity and the Euclidean algorithm for polynomials over a field).

[F5]

The Hermitian norm $∥ \cdot ∥_{2, C}$ is the norm defined in [F1]. Any two norms on the finite-dimensional complex vector space $V_{C}$ are equivalent (All norms on a finite-dimensional complex normed space are equivalent, Equivalent norms, and the dictionary with equivalent metrics): there is $K \geq 1$ such that $K^{- 1} ∥ z ∥_{\infty} \leq ∥ z ∥_{2, C} \leq K ∥ z ∥_{\infty}$ for every $z \in V_{C}$ , where $∥ \cdot ∥_{\infty}$ is the maximum norm of the basis fixed in step 2.3. In particular, $∥ ι v ∥_{2, C} = ∥ v ∥_{2}$ .

Proof

technique · direct; split the spectrum into left and right half-planes, build the corresponding real invariant subspaces from coprime factors of the minimal polynomial, and estimate each Jordan block by its exponential series

1.1givenF2F4

Let $Λ$ be the set of eigenvalues of $T_{C}$ and, for $λ \in Λ$ , let $e_{λ} \geq 1$ be its multiplicity as a root of the real polynomial $μ_{T}$ ; by [F2] all elements of $Λ$ have nonzero real part, and $Λ$ is stable under conjugation with $e_{\overset{ˉ}{λ}} = e_{λ}$ : conjugating $μ_{T} = (x - λ)^{e_{λ}} q$ with $q (λ) \neq 0$ gives $μ_{T} = (x - \overset{ˉ}{λ})^{e_{λ}} \overset{ˉ}{q}$ with $\overset{ˉ}{q} (\overset{ˉ}{λ}) \neq 0$ . Define $p_{s} : = \prod_{Re λ < 0} (x - λ)^{e_{λ}}, p_{u} : = \prod_{Re λ > 0} (x - λ)^{e_{λ}} .$ Conjugation invariance of $Λ$ with equal multiplicities shows $p_{s}, p_{u} \in R [x]$ ; by construction $μ_{T} = p_{s} p_{u}$ , and $p_{s}, p_{u}$ have no common complex root, hence are coprime with a Bézout identity $a p_{s} + b p_{u} = 1$ over $R [x]$ [F4].

1.2F1F3algebra

Exponential on a generalised eigenspace: fix $λ \in Λ$ and put $A_{λ} : = λ I + N_{λ}$ on $G_{λ}$ with $N_{λ}^{e_{λ}} = 0$ as in [F3]. For the partial sum through $m = M$ , the binomial expansion and nilpotence of $N_{λ}$ give the finite rearrangement $\sum_{m = 0}^{M} \frac{t^{m}}{m!} A_{λ}^{m} = \sum_{k = 0}^{e_{λ} - 1} \frac{t^{k}}{k!} N_{λ}^{k} \sum_{j = 0}^{M - k} \frac{(t λ)^{j}}{j!},$ with an empty inner sum when $M < k$ . As $M \to \infty$ , the left side converges to $e^{t A_{λ}}$ by [F1], and each of the finitely many inner sums converges to $e^{t λ}$ by $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ . Thus $e^{t A_{λ}} = \sum_{k = 0}^{e_{λ} - 1} \frac{t^{k}}{k!} N_{λ}^{k} e^{t λ} .$

2.1step 1.1F2F4algebra

Define $E_{s} : = \ker p_{s} (T)$ and $E_{u} : = \ker p_{u} (T)$ . Both are $T$ -invariant because $T$ commutes with $p_{s} (T)$ and $p_{u} (T)$ . Every $v \in V$ satisfies $v = (a p_{s} + b p_{u}) (T) v = a (T) p_{s} (T) v + b (T) p_{u} (T) v \in E_{u} + E_{s},$ the membership following from $p_{u} (T) a (T) p_{s} (T) = a (T) μ_{T} (T) = 0$ on the first summand and $p_{s} (T) b (T) p_{u} (T) = b (T) μ_{T} (T) = 0$ on the second, where $μ_{T} (T) = 0$ (The annihilator set $Ann (T) = {p \in F [x] : p (T) = 0}$ ; once existence is proved, its unique monic generator $μ_{T}$ is the minimal polynomial). If $v \in E_{s} \cap E_{u}$ , the same identity gives $v = a (T) p_{s} (T) v + b (T) p_{u} (T) v = 0$ . Hence $V = E_{s} \oplus E_{u}$ .

2.2step 1.1F3F4algebra

Complexified description: $(E_{s})_{C} = \ker p_{s} (T_{C}) = ⨁_{Re λ < 0} G_{λ}$ and $(E_{u})_{C} = \ker p_{u} (T_{C}) = ⨁_{Re λ > 0} G_{λ}$ . Writing $z = ι x + i ι y$ with $x, y \in V$ , one has $p_{s} (T_{C}) z = ι (p_{s} (T) x) + i ι (p_{s} (T) y)$ , so $\ker p_{s} (T_{C}) = (\ker p_{s} (T))_{C}$ ; the same holds for $p_{u}$ . Indeed, on a summand $G_{λ}$ with $Re λ < 0$ , the factor $(x - λ)^{e_{λ}}$ divides $p_{s}$ , so $p_{s} (T_{C})$ vanishes on $G_{λ}$ ; on a summand with $Re λ > 0$ , $p_{s}$ and $(x - λ)^{e_{λ}}$ are coprime, and evaluating a Bézout identity on $G_{λ}$ (where $(T_{C} - λ)^{e_{λ}} = 0$ ) shows $p_{s} (T_{C})$ is invertible there, so its kernel meets $G_{λ}$ trivially. Summing over [F3] gives the claim; the same argument with $p_{u}$ gives the second identity.

2.3step 1.2F1algebra

Estimates on one generalised eigenspace: fix a basis of $V_{C}$ adapted to the decomposition $V_{C} = ⨁_{λ} G_{λ}$ (for instance the union of Jordan bases of the summands, whose existence is [F3]), and let $∥ \cdot ∥_{\infty}$ be the corresponding maximum norm; then every $z = \sum_{λ} z_{λ}$ has its coordinate block in $G_{λ}$ equal to the coordinate block of $z_{λ}$ , so $∥ z_{λ} ∥_{\infty} \leq ∥ z ∥_{\infty}$ for every $λ$ . On each block use the induced operator norm $∥ N_{λ} ∥_{\infty} : = \sup {∥ N_{λ} z ∥_{\infty} : z \in G_{λ}, ∥ z ∥_{\infty} \leq 1}$ ; this is finite for a finite matrix. Put $P_{λ} (t) : = \sum_{k = 0}^{e_{λ} - 1} \frac{∣ t ∣^{k}}{k!} ∥ N_{λ} ∥_{\infty}^{k},$ so that $∥ e^{t A_{λ}} z ∥_{\infty} \leq e^{(Re λ) t} P_{λ} (t) ∥ z ∥_{\infty}$ for $z \in G_{λ}$ and all real $t$ by step 1.2. Choose $c : = \frac{1}{2} \min_{λ \in Λ} ∣ Re λ ∣ > 0.$ For $u \geq 0$ and $a > 0$ , the nonnegative exponential series gives $e^{a u} \geq (a u)^{k} / k!$ , hence $u^{k} e^{- a u} \leq k! a^{- k}$ for every integer $k \geq 0$ . Applying this to each term of $P_{λ} (\pm u)$ , with $a = - Re λ - c > 0$ on stable blocks and $a = Re λ - c > 0$ on unstable blocks, proves that the following suprema are finite: for $Re λ < 0$ , $C_{λ}^{s} : = \sup_{t \geq 0} P_{λ} (t) e^{(Re λ + c) t} < + \infty,$ and for $Re λ > 0$ , $C_{λ}^{u} : = \sup_{t \leq 0} P_{λ} (t) e^{(Re λ - c) t} < + \infty .$

3.1step 2.22.3F1F5algebra

Constants for the two invariant subspaces. By steps 2.2 and 2.3, for $z \in (E_{s})_{C}$ and $t \geq 0$ the decomposition $z = \sum_{Re λ < 0} z_{λ}$ yields $∥ e^{t T_{C}} z ∥_{\infty} \leq \sum_{Re λ < 0} e^{(Re λ) t} P_{λ} (t) ∥ z_{λ} ∥_{\infty} \leq e^{- c t} (\sum_{Re λ < 0} C_{λ}^{s}) ∥ z ∥_{\infty},$ since, in the adapted basis of step 2.3, $∥ z_{λ} ∥_{\infty} \leq ∥ z ∥_{\infty}$ , and $Re λ + c \leq - c$ for every $Re λ < 0$ (as $c = \frac{1}{2} \min ∣ Re λ ∣$ ). Similarly, for $z \in (E_{u})_{C}$ and $t \leq 0$ , $∥ e^{t T_{C}} z ∥_{\infty} \leq e^{c t} (\sum_{Re λ > 0} C_{λ}^{u}) ∥ z ∥_{\infty} .$ Set $C : = \max (1, K^{2} \sum_{Re λ < 0} C_{λ}^{s}, K^{2} \sum_{Re λ > 0} C_{λ}^{u})$ with $K$ as in [F5]. For $v \in E_{s}$ and $t \geq 0$ , applying the first estimate to $z = ι v$ and using [F5] in both directions, $∥ e^{t T} v ∥_{2} = ∥ e^{t T_{C}} ι v ∥_{2, C} \leq K ∥ e^{t T_{C}} ι v ∥_{\infty} \leq K e^{- c t} (\sum_{Re λ < 0} C_{λ}^{s}) ∥ ι v ∥_{\infty} \leq K^{2} (\sum_{Re λ < 0} C_{λ}^{s}) e^{- c t} ∥ v ∥_{2} \leq C e^{- c t} ∥ v ∥_{2}$ , which is clause 1; the same computation with the second estimate gives clause 2 for $v \in E_{u}$ , $t \leq 0$ .

4.1step 3.1F1algebra

Clause 3: let $u \in E_{u}$ and $t \geq 0$ . By [F1] $e^{t T}$ is invertible with inverse $e^{- t T}$ , and $e^{- t T} u \in E_{u}$ because $E_{u}$ is $T$ -invariant. Applying clause 2 with the time $- t \leq 0$ to the vector $e^{t T} u$ gives $∥ u ∥_{2} = ∥ e^{- t T} e^{t T} u ∥_{2} \leq C e^{- c t} ∥ e^{t T} u ∥_{2},$ hence $∥ e^{t T} u ∥_{2} \geq C^{- 1} e^{c t} ∥ u ∥_{2}$ .

5.1step 3.14.1F1algebra∎

Clause 4: by clauses 1 and 3, every $v \in E_{s}$ satisfies $e^{t T} v \to 0$ as $t \to + \infty$ , and by clause 2 every $v \in E_{u}$ satisfies $e^{t T} v \to 0$ as $t \to - \infty$ . Conversely, let $v \in V$ satisfy $e^{t T} v \to 0$ as $t \to + \infty$ and write $v = u + s$ with $u \in E_{u}$ , $s \in E_{s}$ ; by clauses 3 and 1, $∥ e^{t T} v ∥_{2} \geq ∥ e^{t T} u ∥_{2} - ∥ e^{t T} s ∥_{2} \geq C^{- 1} e^{c t} ∥ u ∥_{2} - C e^{- c t} ∥ s ∥_{2} \to + \infty$ unless $u = 0$ . So convergence to $0$ forces $u = 0$ , that is $v \in E_{s}$ . The characterisation of $E_{u}$ follows by the same argument with $t \to - \infty$ : clause 2 gives $∥ e^{t T} u ∥_{2} \leq C e^{c t} ∥ u ∥_{2} \to 0$ for $u \in E_{u}$ , while for $s \neq 0$ the growth estimate $∥ e^{t T} s ∥_{2} \geq C^{- 1} e^{- c t} ∥ s ∥_{2}$ for $t \leq 0$ , obtained from clause 1 at time $- t \geq 0$ and invertibility of $e^{t T}$ , forces divergence.

Remarks

The subspaces $E_{s}$ and $E_{u}$ are the stable and unstable subspaces of $T$ ; they are constructed from the coprime factorisation $μ_{T} = p_{s} p_{u}$ and are real by construction, so no separate descent argument from the complexification is needed. The constant $C$ depends on the basis fixed in step 2.3, through the equivalence constant $K$ of [F5], and on $T$ , while $c$ is any positive number below $\min_{λ} ∣ Re λ ∣$ .
The hypothesis is a spectral gap at the imaginary axis. A purely imaginary eigenvalue (including the eigenvalue $0$ ) is not covered: if $0$ is an eigenvalue then no positive rate $c$ can work for the corresponding directions. An equilibrium whose linearisation has all eigenvalues in the open left half-plane is locally exponentially stable and A hyperbolic equilibrium whose linearisation has an eigenvalue with positive real part is unstable use only the cases $E_{u} = {0}$ (all eigenvalues in the open left half-plane) and $E_{u} \neq {0}$ .

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

An equilibrium whose linearisation has all eigenvalues in the open left half-plane is locally exponentially stable

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open with $0 \in U$ , and let $F : U \to R^{n}$ be of class $C^{1}$ ( $C^{k}$ Euclidean maps and diffeomorphisms) with $F (0) = 0$ . Put $A : = D F (0)$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case) and suppose that every complex eigenvalue of $A$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism) has negative real part.

Then $0$ is a locally exponentially stable equilibrium of $z^{'} = F (z)$ (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system): there are real numbers $δ > 0$ , $β \geq 1$ and $γ > 0$ such that every maximal solution $z$ of $z^{'} = F (z)$ with $∥ z (0) ∥_{2} < δ$ is defined for all $t \geq 0$ and satisfies

$∥ z (t) ∥_{2} \leq β e^{- γ t} ∥ z (0) ∥_{2} (t \geq 0) .$

In particular $0$ is Lyapunov stable and asymptotically stable in the sense of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system.

Facts & Assumptions

Given: The open set $U$ with $0 \in U$ , the $C^{1}$ field $F$ with $F (0) = 0$ , and the matrix $A = D F (0)$ all of whose eigenvalues have negative real part.

[F1]

The exponential series defines the differentiable family $t \mapsto e^{t A}$ with $\frac{d}{d t} e^{t A} = A e^{t A} = e^{t A} A$ and $e^{0} = I_{n}$ , and exponential dichotomy gives, since no eigenvalue of $A$ has real part $0$ and all eigenvalues have negative real part, constants $C \geq 1$ , $c > 0$ with $∥ e^{t A} v ∥_{2} \leq C e^{- c t} ∥ v ∥_{2}$ for all $v \in R^{n}$ and $t \geq 0$ , with no unstable part. To justify this last deduction explicitly, use the complexified subspaces constructed in the cited dichotomy proof, steps 1.1 and 2.1-2.2: $(E_{u})_{C}$ is the direct sum of the generalised eigenspaces with $Re λ > 0$ . Under the present hypothesis this is the empty direct sum ${0}$ (equivalently the polynomial $p_{u}$ is $1$ , so $E_{u} = \ker I = {0}$ ). Hence $E_{s} = R^{n}$ , and clause 1 of the dichotomy gives the stated estimate on every real vector (The exponential series of a bounded operator, Exponential dichotomy for a hyperbolic endomorphism: invariant stable and unstable subspaces with exponential estimates).

[F2]

Differentiability at $0$ : for every $η > 0$ there is $ρ > 0$ such that $∥ F (z) - A z ∥_{2} \leq η ∥ z ∥_{2}$ whenever $∥ z ∥_{2} < ρ$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[F3]

Improper integrals: For $a > 0$ , direct integration using The exponential function is smooth and $(\exp)^{'} = \exp$ gives $\int_{0}^{T} e^{- a t} d t = (1 - e^{- a T}) / a \to 1 / a$ by The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$ . if a continuous $f : [0, \infty) \to R$ satisfies $∣ f (t) ∣ \leq M e^{- λ t}$ for constants $M, λ > 0$ , then $\int_{0}^{\infty} f (t) d t$ converges absolutely and $\lim_{T \to \infty} \int_{0}^{T} f (t) d t = \int_{0}^{\infty} f (t) d t$ (Improper integrals over unbounded intervals, Comparison tests for improper integrals); the fundamental theorem applies on compact intervals (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F4]

Compact continuation: an autonomous solution that is trapped in a compact subset of its open state domain near a finite endpoint extends past that endpoint (Compact continuation for locally Lipschitz ODEs: a solution whose graph stays in a compact subset of the domain near a finite endpoint extends past it); $F$ is locally Lipschitz because it is $C^{1}$ .

[F5]

The Euclidean norm is induced by $⟨ \cdot, \cdot ⟩_{2}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ) and satisfies Cauchy--Schwarz (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation). For a matrix $Q$ , use the induced operator norm $∥ Q ∥_{2} : = \sup {∥ Q z ∥_{2} : ∥ z ∥_{2} \leq 1}$ . By the real spectral theorem, a symmetric positive definite matrix $Q$ has an orthonormal eigenbasis with positive eigenvalues; writing $λ_{\min} (Q)$ and $λ_{\max} (Q)$ for their minimum and maximum gives $λ_{\min} (Q) ∥ z ∥_{2}^{2} \leq z^{T} Q z \leq λ_{\max} (Q) ∥ z ∥_{2}^{2}$ and $∥ Q ∥_{2} = λ_{\max} (Q)$ (diagonalize in that basis and use the Euclidean norm) (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis). Balls are as in Open ball, closed ball and sphere in a metric space, and the chain rule computes the derivative of a quadratic form along a curve (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F6]

Closed and bounded subsets of $R^{n}$ are compact for the Euclidean metric (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

Proof

technique · direct; build the quadratic Lyapunov function from the exponential dichotomy and compare its decay with the norm

1.1F1F3F5algebra

Define the symmetric matrix $Q$ entrywise by $Q_{i j} : = \int_{0}^{\infty} (e^{t A^{T}} e^{t A})_{i j} d t .$ Each integrand is continuous and, for standard basis vectors $e_{i}, e_{j}$ , equals $⟨ e^{t A} e_{i}, e^{t A} e_{j} ⟩_{2}$ ; Cauchy--Schwarz and [F1], [F5] bound its absolute value by $C^{2} e^{- 2 c t}$ . Hence every entry integral converges absolutely by [F3]. The resulting matrix is symmetric because $e^{t A^{T}} e^{t A}$ is symmetric for every $t$ .

2.1step 1.1F1F3F5algebra

The matrix $Q$ is positive definite and satisfies the Lyapunov equation $A^{T} Q + Q A = - I_{n}$ . Positivity: for $z \neq 0$ , $z^{T} Q z = \int_{0}^{\infty} ∥ e^{t A} z ∥_{2}^{2} d t$ , the integrand is continuous, nonnegative, and strictly positive on a neighbourhood of $t = 0$ , so the integral is $> 0$ ; for $z = 0$ it is $0$ . Lyapunov equation: put $Q_{T} : = \int_{0}^{T} e^{t A^{T}} e^{t A} d t$ . By [F3] and linearity of the integral on compact intervals, $A^{T} Q_{T} + Q_{T} A = \int_{0}^{T} \frac{d}{d t} (e^{t A^{T}} e^{t A}) d t = e^{T A^{T}} e^{T A} - I_{n}$ , by the product rule and $\frac{d}{d t} e^{t A} = A e^{t A} = e^{t A} A$ from [F1]. Letting $T \to \infty$ , the left side tends to $A^{T} Q + Q A$ by [F3]. Each entry of $e^{T A^{T}} e^{T A}$ equals $⟨ e^{T A} e_{i}, e^{T A} e_{j} ⟩_{2}$ for standard basis vectors $e_{i}, e_{j}$ , so [F1], [F5] bound it in absolute value by $C^{2} e^{- 2 c T} \to 0$ . Hence $A^{T} Q + Q A = - I_{n}$ .

3.1step 2.1F2F5F6algebra

Fix $η > 0$ with $2 ∥ Q ∥_{2} η \leq \frac{1}{2}$ and let $ρ > 0$ be a radius as in [F2] for this $η$ ; since $U$ is open and $0 \in U$ , shrink $ρ$ if necessary so that $\overline{B} (0, ρ) \subseteq U$ (the implication of [F2] persists under shrinking $ρ$ ; balls as in Open ball, closed ball and sphere in a metric space). Define $V (z) : = z^{T} Q z$ ; since $Q$ is positive definite with smallest eigenvalue $λ_{\min} (Q) > 0$ , choose $0 < α < λ_{\min} (Q) ρ^{2}$ . Then $S : = {z \in U : V (z) \leq α}$ satisfies $S \subseteq B (0, ρ) \subseteq \overline{B} (0, ρ) \subseteq U$ , so $S = {z \in \overline{B} (0, ρ) : V (z) \leq α}$ is closed and bounded, hence compact by [F6].

4.1step 3.1F2F5algebra

Let $z$ be a solution of $z^{'} = F (z)$ . On the interval where $∥ z (t) ∥_{2} < ρ$ , $\frac{d}{d t} V (z (t)) = 2 z (t)^{T} Q z^{'} (t) = z^{T} (A^{T} Q + Q A) z + 2 z^{T} Q (F (z) - A z) \leq - ∥ z ∥_{2}^{2} + 2 η ∥ Q ∥_{2} ∥ z ∥_{2}^{2} \leq - \frac{1}{2} ∥ z ∥_{2}^{2},$ by step 2.1, [F2] and the choice of $η$ ; the derivative exists by the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , [F5]). Since $V (z) \leq λ_{\max} (Q) ∥ z ∥_{2}^{2}$ , this gives $\frac{d}{d t} V (z (t)) \leq - \frac{1}{2 λ_{\max} (Q)} V (z (t))$ while the solution stays in $B (0, ρ)$ .

5.1step 3.14.1F4

A maximal solution with $z (0) \in S$ stays in $S$ and is defined for all $t \geq 0$ . Indeed, while it stays in $S \subseteq B (0, ρ)$ , step 4.1 shows that $t \mapsto V (z (t))$ is nonincreasing, so it cannot leave $S$ through the level $V = α$ ; it cannot leave $S$ through $\partial U$ either, because $S$ is compact in $U$ . If its maximal forward interval were $[0, b)$ with $b < \infty$ , the solution would be trapped in the compact set $S \subseteq U$ near $b$ , so [F4] would extend it, a contradiction. Hence it is defined for all $t \geq 0$ and stays in $S$ ; this positive invariance of $S$ is the property used to verify the stability definition of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system.

6.1step 4.15.1algebra∎

Exponential decay: by step 4.1, $\frac{d}{d t} V (z (t)) \leq - γ_{V} V (z (t))$ for all $t \geq 0$ whenever the solution stays in $S \subseteq B (0, ρ)$ , with $γ_{V} : = \frac{1}{2 λ_{\max} (Q)} > 0$ . A function with nonpositive derivative of $t \mapsto e^{γ_{V} t} V (z (t))$ is nonincreasing, so $V (z (t)) \leq e^{- γ_{V} t} V (z (0))$ . Combining with $λ_{\min} (Q) ∥ z ∥_{2}^{2} \leq V (z) \leq λ_{\max} (Q) ∥ z ∥_{2}^{2}$ gives $∥ z (t) ∥_{2} \leq β e^{- γ_{V} t / 2} ∥ z (0) ∥_{2}, β : = \sqrt{λ_{\max} (Q) / λ_{\min} (Q)} \geq 1,$ for solutions with $z (0) \in S$ . Choose $δ > 0$ with $B (0, δ) \subseteq S$ ; then every maximal solution with $∥ z (0) ∥_{2} < δ$ satisfies the displayed estimate. With $γ : = γ_{V} / 2 > 0$ this proves the exponential bound, and it implies Lyapunov stability and asymptotic stability directly: given $ε > 0$ small, choosing initial data with $∥ z (0) ∥_{2} < δ^{'}$ where $δ^{'} \leq δ$ and $β δ^{'} < ε$ keeps $∥ z (t) ∥_{2} < ε$ for all $t \geq 0$ , and the bound gives convergence to $0$ .

Remarks

The quadratic form $Q$ is the classical Lyapunov matrix $\int_{0}^{\infty} e^{t A^{T}} e^{t A} d t$ , positive definite exactly because all eigenvalues of $A$ lie in the open left half-plane. The proof uses the exponential dichotomy only through the estimate of [F1]; hyperbolicity with a nonempty unstable part would require the indefinite quadratic form of A hyperbolic equilibrium whose linearisation has an eigenvalue with positive real part is unstable instead.
The conclusion is local: the constants $δ, β, γ$ depend on the field $F$ and are not uniform over an arbitrary family of systems. The theorem does not assert a global domain of attraction, an optimal rate, or any nonlinear analogue of the eigenvalue criterion when eigenvalues lie on the imaginary axis.

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A hyperbolic equilibrium whose linearisation has an eigenvalue with positive real part is unstable

Statement

no complex eigenvalue of $A$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism) has real part $0$ , and
at least one eigenvalue of $A$ has positive real part.

Then the equilibrium $0$ is not Lyapunov stable (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system): there is a real number $ε_{0} > 0$ such that for every $δ > 0$ there exist $z_{0} \in U$ with $∥ z_{0} ∥_{2} < δ$ and a time $t_{1} > 0$ such that the maximal solution $z$ of $z^{'} = F (z)$ , $z (0) = z_{0}$ is defined on $[0, t_{1}]$ and satisfies

$∥ z (t_{1}) ∥_{2} = ε_{0} .$

Facts & Assumptions

Given: The open set $U$ with $0 \in U$ , the $C^{1}$ field $F$ with $F (0) = 0$ , and the matrix $A = D F (0)$ with no purely imaginary eigenvalue and at least one eigenvalue of positive real part.

[F1]

The complexification of $A$ is $A_{C} : = {id}_{C} \otimes A$ on $C \otimes_{R} R^{n}$ (Complexification as $C \otimes_{R} V$ with its canonical real-linear embedding, Complexification of a real-linear map). In the standard basis write a complex vector as $w = p + i q$ with $p, q \in R^{n}$ and equip it with $∥ p + i q ∥_{2, C} : = (∥ p ∥_{2}^{2} + ∥ q ∥_{2}^{2})^{1 / 2}$ . The real and imaginary coordinate identification makes this a complete normed space ( $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ , $C$ is the real coordinate plane, with coordinate arithmetic), and the finite real matrix of $A_{C}$ is bounded by Cauchy--Schwarz (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation). If $λ$ is an eigenvalue of $A_{C}$ with $Re λ > 0$ , choose $w = p + i q \neq 0$ with $A_{C} w = λ w$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism). The exponential series gives $e^{t A_{C}} w = e^{t λ} w$ , and $∣ e^{t λ} ∣ = e^{t Re λ} \to 0$ as $t \to - \infty$ (The exponential series of a bounded operator, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ). Since $A$ is real, the real and imaginary parts of $e^{t A_{C}} w$ are $e^{t A} p$ and $e^{t A} q$ ; each tends to $0$ in the Euclidean norm. At least one of $p, q$ is nonzero. Exponential dichotomy gives $A$ -invariant subspaces $R^{n} = E_{s} \oplus E_{u}$ and constants $C \geq 1$ , $c > 0$ such that $∥ e^{t A} v ∥_{2} \leq C e^{- c t} ∥ v ∥_{2}$ for $v \in E_{s}$ , $t \geq 0$ , $∥ e^{t A} v ∥_{2} \leq C e^{c t} ∥ v ∥_{2}$ for $v \in E_{u}$ , $t \leq 0$ , and $∥ e^{t A} v ∥_{2} \geq C^{- 1} e^{c t} ∥ v ∥_{2}$ for $v \in E_{u}$ , $t \geq 0$ ; moreover $E_{s}, E_{u}$ are characterised by convergence to $0$ at $+ \infty$ and $- \infty$ respectively (Exponential dichotomy for a hyperbolic endomorphism: invariant stable and unstable subspaces with exponential estimates). Clause 4 places a nonzero real vector among $p, q$ that tends to $0$ as $t \to - \infty$ in $E_{u}$ . Thus $E_{u} \neq {0}$ .

[F2]

Differentiability at $0$ : for every $η > 0$ there is $ρ > 0$ with $∥ F (z) - A z ∥_{2} \leq η ∥ z ∥_{2}$ for $∥ z ∥_{2} < ρ$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[F3]

Let $Π_{u}, Π_{s}$ be the projections onto $E_{u}, E_{s}$ along the decomposition; they are linear and satisfy $Π_{u} A = A Π_{u}$ , $Π_{s} A = A Π_{s}$ , $Π_{u} + Π_{s} = I$ . For a linear map $L$ between these Euclidean spaces, write $∥ L ∥_{2} : = \sup {∥ L v ∥_{2} : ∥ v ∥_{2} \leq 1}$ for its induced operator norm. In orthonormal coordinates every linear map has a finite matrix, and Cauchy--Schwarz bounds its operator norm by the square root of the sum of squares of its entries; hence $K : = \max (∥ Π_{u} ∥_{2}, ∥ Π_{s} ∥_{2}) < \infty$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F4]

Improper integrals For $a > 0$ , direct integration using The exponential function is smooth and $(\exp)^{'} = \exp$ gives $\int_{0}^{T} e^{- a t} d t = (1 - e^{- a T}) / a \to 1 / a$ by The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$ . Improper integrals of continuous functions with exponential decay converge absolutely, by comparison (Improper integrals over unbounded intervals, Comparison tests for improper integrals); Cauchy--Schwarz bounds Euclidean inner products by the product of norms (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F5]

Compact continuation: an autonomous solution trapped in a compact subset of its open state domain near a finite endpoint extends past that endpoint; $F$ is locally Lipschitz because it is $C^{1}$ (Compact continuation for locally Lipschitz ODEs: a solution whose graph stays in a compact subset of the domain near a finite endpoint extends past it). Closed Euclidean balls are compact because they are closed and bounded (Open ball, closed ball and sphere in a metric space, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F6]

If $Q$ is a symmetric positive definite endomorphism of a finite-dimensional real inner product space, the real spectral theorem gives an orthonormal eigenbasis with positive eigenvalues; writing $λ_{\max} (Q)$ for the largest eigenvalue yields $⟨ Q v, v ⟩ \leq λ_{\max} (Q) ∥ v ∥_{2}^{2}$ (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

Proof

technique · direct; build an indefinite quadratic Lyapunov function that grows in the unstable directions and use exponential dichotomy estimates

1.1F1F4algebra

Choose a nonzero $u_{0} \in E_{u}$ , which is possible because $E_{u} \neq {0}$ by [F1]. On $E_{u}$ define $Q_{u} : = \int_{0}^{\infty} e^{- t A_{u}^{T}} e^{- t A_{u}} d t, A_{u} : = A ∣_{E_{u}},$ and on $E_{s}$ define $Q_{s} : = \int_{0}^{\infty} e^{t A_{s}^{T}} e^{t A_{s}} d t$ , $A_{s} : = A ∣_{E_{s}}$ ; the transposes are with respect to the inherited Euclidean inner products. If $E_{s} = {0}$ , $Q_{s}$ is the zero endomorphism of that space and its positivity assertion is vacuous. By [F1], $∥ e^{- t A_{u}} v ∥_{2} \leq C e^{- c t} ∥ v ∥_{2}$ for $v \in E_{u}$ , $t \geq 0$ , and $∥ e^{t A_{s}} v ∥_{2} \leq C e^{- c t} ∥ v ∥_{2}$ for $v \in E_{s}$ , $t \geq 0$ . Cauchy--Schwarz [F4] bounds each matrix entry of both integrands by $C^{2} e^{- 2 c t}$ , so [F4] gives absolute convergence entrywise. For nonzero $v$ in either nonzero subspace, the corresponding quadratic form equals the integral of $∥ e^{- t A_{u}} v ∥_{2}^{2}$ or $∥ e^{t A_{s}} v ∥_{2}^{2}$ ; the integrand is continuous and positive at $t = 0$ , so $Q_{u}$ and $Q_{s}$ are positive definite on their subspaces. For $Q_{u, T} : = \int_{0}^{T} e^{- t A_{u}^{T}} e^{- t A_{u}} d t$ , differentiating the integrand and applying the fundamental theorem gives $A_{u}^{T} Q_{u, T} + Q_{u, T} A_{u} = I - e^{- T A_{u}^{T}} e^{- T A_{u}}$ ; the terminal term tends to $0$ by the decay estimate, so $A_{u}^{T} Q_{u} + Q_{u} A_{u} = I_{E_{u}}$ . The same calculation with $Q_{s, T} : = \int_{0}^{T} e^{t A_{s}^{T}} e^{t A_{s}} d t$ gives $A_{s}^{T} Q_{s} + Q_{s} A_{s} = - I_{E_{s}}$ after taking $T \to \infty$ .

2.1step 1.1F2F3F4algebra

Define the function $V$ on $U$ by $V (z) : = ⟨ Q_{u} Π_{u} z, Π_{u} z ⟩_{2} - ⟨ Q_{s} Π_{s} z, Π_{s} z ⟩_{2}$ , writing $u = Π_{u} z$ , $s = Π_{s} z$ . It is a quadratic form, hence differentiable, and for a solution $z$ of $z^{'} = F (z)$ with $z (t) = u (t) + s (t)$ one has $u^{'} = A_{u} u + Π_{u} r (z)$ , $s^{'} = A_{s} s + Π_{s} r (z)$ where $r (z) : = F (z) - A z$ , because $Π_{u}, Π_{s}$ commute with $A$ [F3]. Differentiating along the solution, $\frac{d}{d t} V (z (t)) = ∥ u ∥_{2}^{2} + ∥ s ∥_{2}^{2} + 2 ⟨ Q_{u} u, Π_{u} r (z) ⟩_{2} - 2 ⟨ Q_{s} s, Π_{s} r (z) ⟩_{2} \geq ∥ u ∥_{2}^{2} + ∥ s ∥_{2}^{2} - 2 M ∥ r (z) ∥_{2} ∥ z ∥_{2},$ with $M : = ∥ Q_{u} ∥_{2} ∥ Π_{u} ∥_{2}^{2} + ∥ Q_{s} ∥_{2} ∥ Π_{s} ∥_{2}^{2} > 0$ : the two scalar-product terms are estimated by Cauchy--Schwarz and $∥ u ∥_{2} \leq ∥ Π_{u} ∥_{2} ∥ z ∥_{2}$ , $∥ s ∥_{2} \leq ∥ Π_{s} ∥_{2} ∥ z ∥_{2}$ ; no orthogonality of the splitting is assumed, and $∥ z ∥_{2} = ∥ u + s ∥_{2} \leq ∥ u ∥_{2} + ∥ s ∥_{2}$ gives $∥ z ∥_{2}^{2} \leq 2 (∥ u ∥_{2}^{2} + ∥ s ∥_{2}^{2})$ .

3.1step 1.12.1F2F3F6algebra

Choose $η > 0$ with $8 M η \leq 1$ and let $ρ > 0$ be as in [F2] for this $η$ ; shrink $ρ$ so that $\overline{B} (0, ρ) \subseteq U$ . For $∥ z ∥_{2} < ρ$ , step 2.1 and [F2] give, since $∥ z ∥_{2}^{2} \leq 2 (∥ u ∥_{2}^{2} + ∥ s ∥_{2}^{2})$ and $8 M η \leq 1$ , $\frac{d}{d t} V (z (t)) \geq ∥ u ∥_{2}^{2} + ∥ s ∥_{2}^{2} - 2 M η ∥ z ∥_{2}^{2} \geq \frac{1}{2} (∥ u ∥_{2}^{2} + ∥ s ∥_{2}^{2}) \geq 0.$ In particular, whenever $V (z) > 0$ then $u \neq 0$ and $∥ u ∥_{2}^{2} \geq V (z) / λ_{\max} (Q_{u})$ by [F6], so $\frac{d}{d t} V (z (t)) \geq \frac{1}{2} ∥ u ∥_{2}^{2} \geq γ V (z (t)), γ : = \frac{1}{2 λ_{\max} (Q_{u})} > 0.$ Set $ε_{0} : = ρ$ .

4.1step 1.13.1

Let $0 < δ$ be arbitrary and choose $z_{0} : = u_{0} \in E_{u}$ with $0 < ∥ u_{0} ∥_{2} < \min (δ, ε_{0})$ (step 1.1). Then $V (z_{0}) = ⟨ Q_{u} u_{0}, u_{0} ⟩_{2} > 0$ . Let $z : [0, b) \to U$ be the maximal forward solution through $z_{0}$ . As long as $∥ z (t) ∥_{2} < ε_{0}$ , step 3.1 gives $\frac{d}{d t} V (z (t)) \geq 0$ whenever $V (z (t)) \geq 0$ ; since $V (z (0)) > 0$ , the function $V \circ z$ stays positive and satisfies $V (z (t)) \geq e^{γ t} V (z_{0})$ on that interval.

5.1step 4.1F5F6algebra∎

The solution must reach the sphere $∥ z ∥_{2} = ε_{0}$ at some finite time. Suppose first that the solution is defined for all $t \geq 0$ and never reaches that sphere. Then $∥ z (t) ∥_{2} < ε_{0}$ for all $t$ , so step 4.1 gives $V (z (t)) \geq e^{γ t} V (z_{0}) \to + \infty$ , while $V (z) \leq λ_{\max} (Q_{u}) ∥ Π_{u} ∥_{2}^{2} ε_{0}^{2}$ on $\overline{B} (0, ε_{0})$ by [F6]; this is a contradiction. Suppose next that the maximal forward interval is $[0, b)$ with $b < \infty$ and the solution never reaches the sphere. Then its values lie in the compact set $\overline{B} (0, ε_{0}) \subseteq U$ near the endpoint $b$ by [F5], so [F5] extends the solution past $b$ , contradicting maximality. Hence in every case there is a first time $t_{1} \in (0, \infty)$ with $∥ z (t_{1}) ∥_{2} = ε_{0}$ , and the solution is defined on $[0, t_{1}]$ . Since $δ > 0$ was arbitrary, the equilibrium $0$ is not Lyapunov stable in the sense of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system.

Remarks

The proof uses the dichotomy hypothesis to obtain the invariant splitting with an exponential gap. If $A$ has an eigenvalue with positive real part and also a purely imaginary eigenvalue, the conclusion remains true in the classical linearisation theorem, but a different argument is needed (the centre directions require separate treatment); that case is not covered here.
The indefinite quadratic form $V$ is unbounded above, so $V$ growing is not by itself a contradiction; the contradiction in step 5.1 comes from $V$ growing while the solution stays in the bounded ball. The exit time $t_{1}$ depends on $z_{0}$ and $δ$ ; the theorem asserts instability, not a uniform escape rate for all small data.

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The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart

Definition

Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be open, let $M : Ω \to M_{n} (R)$ be of class $C^{2}$ ( $C^{k}$ Euclidean maps and diffeomorphisms) with $M (q)$ symmetric and positive definite for every $q \in Ω$ , that is $v^{T} M (q) v > 0$ for every nonzero $v \in R^{n}$ , and let $U : Ω \to R$ be of class $C^{2}$ . The matrix field $M$ is the kinetic metric field and $U$ the potential of the data in the chart $Ω$ .

Coordinates and matrix entries are indexed by $0 \leq i, j, k < n$ , as in The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. Vectors are read as columns: for $A \in M_{n} (R)$ , $(A v)_{i} : = \sum_{j = 0}^{n - 1} A_{i j} v_{j}$ , and $v^{T} A v : = \sum_{i, j = 0}^{n - 1} v_{i} A_{i j} v_{j}$ .

Define, for $(q, v) \in Ω \times R^{n}$ , the kinetic energy, Lagrangian and energy

$T (q, v) : = \frac{1}{2} v^{T} M (q) v, L (q, v) : = T (q, v) - U (q), E (q, v) : = T (q, v) + U (q),$

and, for $q \in Ω$ and $v \in R^{n}$ , the matrix

$D_{q} M (q) [v] : = \sum_{k = 0}^{n - 1} v_{k} \partial_{k} M (q), (\partial_{k} M (q))_{i j} = \partial_{k} M_{i j} (q) (0 \leq i, j, k < n) .$

The mechanical vector field of the data is the map

$G : Ω \times R^{n} ⟶ R^{n} \times R^{n}, G (q, v) = (v, M (q)^{- 1} (\frac{1}{2} \nabla_{q} (v^{T} M (q) v) - (D_{q} M (q) [v]) v - \nabla U (q))),$

where $\nabla_{q}$ and $\nabla U$ are Euclidean gradients (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case) and $M (q)^{- 1}$ is the inverse of the positive definite matrix $M (q)$ (Invertible matrices and the general linear group ${GL}_{n} (F)$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes). The mechanical system in the chart is the first-order system

$q^{'} (t) = v (t), v^{'} (t) = M (q (t))^{- 1} (\frac{1}{2} \nabla_{q} f (q (t), v (t)) - (D_{q} M (q (t)) [v (t)]) v (t) - \nabla U (q (t))), f (q, v) : = v^{T} M (q) v .$

A point $q_{*} \in Ω$ is a mechanical equilibrium when $\nabla U (q_{*}) = 0$ .

Remarks

Equilibria. If $q_{*} \in Ω$ is a mechanical equilibrium, then $G (q_{*}, 0) = (0, 0)$ : the first component is $0$ , and in the second component $\nabla_{q} f (q_{*}, 0) = 0$ (a quadratic form in $v$ evaluated at $v = 0$ ), $(D_{q} M (q_{*}) [0]) 0 = 0$ and $\nabla U (q_{*}) = 0$ . Thus $(q_{*}, 0)$ is an equilibrium of $G$ in the sense of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system.
Equivalence with the Euler-Lagrange equation. A $C^{2}$ curve $q : I \to Ω$ with velocity $v : = q^{'}$ satisfies the Euler-Lagrange equation $\frac{d}{d t} \nabla_{v} L = \nabla_{q} L$ of the Lagrangian above if and only if $(q, v)$ solves the system $q^{'} = v$ , $v^{'} = G_{2} (q, v)$ . Indeed $\nabla_{v} L = M (q) v$ , so

$\frac{d}{d t} (M (q) q^{'}) = M (q) q^{''} + (D_{q} M (q) [q^{'}]) q^{'}, \nabla_{q} L = \frac{1}{2} \nabla_{q} (q^{' T} M (q) q^{'}) - \nabla U (q),$

and multiplying the Euler-Lagrange equation by $M (q)^{- 1}$ gives the displayed second component. The computation is linear algebra plus the product rule for the matrix-valued curve $t \mapsto M (q (t))$ .

Regularity. Under the stated hypotheses the map $G$ is of class $C^{1}$ on $Ω \times R^{n}$ : $M$ is $C^{2}$ , so the maps involving $\partial_{k} M$ , $\nabla_{q} f$ and $D_{q} M [\cdot] \cdot$ are $C^{1}$ ; $\nabla U$ is $C^{1}$ because $U$ is $C^{2}$ ; and $q \mapsto M (q)^{- 1}$ is $C^{1}$ wherever $\det M (q) \neq 0$ , since $M^{- 1} = \det (M)^{- 1} a d j (M)$ has entries that are polynomial in the entries of $M$ divided by the continuous nonvanishing function $\det M$ . Hence the mechanical system satisfies the Picard-Lindelöf regularity hypotheses locally.
Scope. This is a mathematical chart model: $M$ is an arbitrary $C^{2}$ positive definite symmetric matrix field and no physical law, identification with masses, units or empirical assertion is made by the definition. The constant case $M (q) \equiv M$ reduces to $M q^{''} = - \nabla U (q)$ , and a mechanical equilibrium is a critical point of the potential.

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Energy is conserved along solutions of the mechanical system with a configuration-dependent kinetic metric

Statement

Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be open, let $M : Ω \to M_{n} (R)$ be a $C^{2}$ field of symmetric positive definite matrices, let $U : Ω \to R$ be $C^{2}$ , and let $E (q, v) = \frac{1}{2} v^{T} M (q) v + U (q)$ be the energy of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart. If $q : I \to Ω$ is a $C^{2}$ solution of the mechanical system $M (q) q^{''} = \frac{1}{2} \nabla_{q} (q^{' T} M (q) q^{'}) - (D_{q} M (q) [q^{'}]) q^{'} - \nabla U (q)$ on the interval $I$ , then

$\frac{d}{d t} E (q (t), q^{'} (t)) = 0 (t \in I);$

that is, the energy is constant along every solution.

Facts & Assumptions

Given: The kinetic metric field $M$ , the potential $U$ , and a $C^{2}$ solution $q$ of the mechanical system, with $v : = q^{'}$ .

[F1]

The mechanical system is equivalent to the Euler-Lagrange equation of $L = T - U$ and reads $M (q) q^{''} = \frac{1}{2} g (q, v) - (D_{q} M (q) [v]) v - \nabla U (q)$ with $g (q, v) : = \nabla_{q} (v^{T} M (q) v)$ and $D_{q} M (q) [v] = \sum_{k} v_{k} \partial_{k} M (q)$ (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart).

[F2]

Chain rule and product rule: for differentiable curves and differentiable functions the usual chain and product rules hold; finite sums may be differentiated termwise, and a $C^{1}$ map of several variables is totally differentiable (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Laws of finite sums and finite products, $C^{k}$ Euclidean maps and diffeomorphisms, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[F3]

Gradient and inner product: $\nabla_{q} (v^{T} M (q) v)$ has components $\partial_{k} (v^{T} M (q) v) = v^{T} \partial_{k} M (q) v$ , so that

$(\nabla_{q} (v^{T} M (q) v)) \cdot v = \sum_{k} v_{k} v^{T} \partial_{k} M (q) v = v^{T} (\sum_{k} v_{k} \partial_{k} M (q)) v = v^{T} (D_{q} M (q) [v]) v$

(The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Laws of finite sums and finite products, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[F4]

A continuous scalar function on an order-convex interval whose derivative vanishes at interior points is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

Proof

technique · direct; differentiate the energy along the solution and substitute the equation of motion

1.1givenF1F2

The map $t \mapsto M (q (t))$ is $C^{1}$ with derivative $\frac{d}{d t} M (q (t)) = \sum_{k} q_{k}^{'} (t) \partial_{k} M (q (t)) = D_{q} M (q (t)) [v (t)]$ , by the chain rule applied to each entry and the definition of $D_{q} M [q^{'}]$ [F1, F2].

1.2F2F3

Differentiating the potential: $\frac{d}{d t} U (q (t)) = \nabla U (q (t)) \cdot v (t)$ by the chain rule.

2.1step 1.1F1F2algebra

Differentiating the kinetic energy: $\frac{d}{d t} (\frac{1}{2} v (t)^{T} M (q (t)) v (t)) = v^{T} M (q) v^{'} + \frac{1}{2} v^{T} (D_{q} M (q) [v]) v,$ using the product rule, symmetry of $M (q)$ , and step 1.1; here and below all functions are evaluated at $t$ .

3.1step 1.12.1F1algebra

Substituting the equation of motion $v^{'} = M (q)^{- 1} (\frac{1}{2} \nabla_{q} (v^{T} M (q) v) - (D_{q} M (q) [v]) v - \nabla U (q))$ of [F1] into step 2.1 gives $\frac{d}{d t} (\frac{1}{2} v^{T} M (q) v) = \frac{1}{2} v^{T} \nabla_{q} (v^{T} M (q) v) - v^{T} (D_{q} M (q) [v]) v - v^{T} \nabla U (q) + \frac{1}{2} v^{T} (D_{q} M (q) [v]) v .$

4.1step 2.11.23.1F3F4algebra∎

By [F3] the first and the last terms combine as $\frac{1}{2} v^{T} \nabla_{q} (v^{T} M (q) v) + \frac{1}{2} v^{T} (D_{q} M [v]) v = v^{T} (D_{q} M [v]) v$ , cancelling the middle term; hence $\frac{d}{d t} (\frac{1}{2} v^{T} M (q) v) = - v^{T} \nabla U (q)$ . Adding step 1.2, $\frac{d}{d t} E (q (t), v (t)) = - v^{T} \nabla U (q) + v^{T} \nabla U (q) = 0$ on $I$ . The continuous scalar energy is constant on the interval $I$ by [F4].

Remarks

The cancellation uses exactly the two terms produced by the configuration dependence of $M$ : the derivative of the kinetic energy along the flow produces $v^{T} (D_{q} M [v]) v$ , and substitution of the two configuration-dependent acceleration terms gives $\frac{1}{2} v^{T} \nabla_{q} (v^{T} M v) - v^{T} (D_{q} M [v]) v = - \frac{1}{2} v^{T} (D_{q} M [v]) v$ , cancelling the kinetic derivative term $\frac{1}{2} v^{T} (D_{q} M [v]) v$ . For constant $M$ the statement reduces to the familiar $\frac{d}{d t} (\frac{1}{2} v^{T} M v + U) = 0$ .
The lemma is a statement about the model: it does not assert that any physical system conserves energy, nor that solutions exist globally, nor that the energy is a Liapunov function for any equilibrium; those uses are made in A strict local minimum of the potential gives a Lyapunov stable equilibrium of the mechanical system, by energy conservation under their own hypotheses.

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Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity

Statement

Let $n \geq 1$ and let $M \in M_{n} (R)$ be symmetric and positive definite, that is $v^{T} M v > 0$ for every nonzero $v \in R^{n}$ ; identify $R^{n}$ with its standard real inner product space (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ). Then:

$M$ is invertible and $M^{- 1}$ is symmetric and positive definite;
there is a unique symmetric positive definite matrix $M^{1 / 2}$ with $(M^{1 / 2})^{2} = M$ ;
$M^{- 1 / 2} : = (M^{1 / 2})^{- 1}$ is symmetric and positive definite, satisfies $(M^{- 1 / 2})^{2} = M^{- 1}$ , and $M^{- 1 / 2} M M^{- 1 / 2} = I_{n}$ ;
for every symmetric matrix $K$ that is non-negative, that is $v^{T} K v \geq 0$ for all $v$ , the matrix $S : = M^{- 1 / 2} K M^{- 1 / 2}$ is symmetric and non-negative, and $S$ is positive definite if and only if $K$ is.

Facts & Assumptions

Given: The symmetric positive definite matrix $M$ and, in clause 4, a symmetric non-negative matrix $K$ .

[F1]

A real matrix is symmetric if and only if the associated endomorphism of $R^{n}$ is self-adjoint; positive definite (respectively non-negative) matrices in the form sense correspond exactly to positive (respectively non-negative) operators, and a positive operator is non-negative (In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose, Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space, Over the reals, non-negative and positive operators correspond exactly to positive semidefinite and positive definite symmetric forms, Non-negative and positive operators).

[F2]

A non-negative operator on a finite-dimensional real inner product space has a unique non-negative square root, and a non-negative operator admits an orthonormal basis of eigenvectors with non-negative eigenvalues (A non-negative operator has a unique non-negative square root, A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S).

[F3]

Matrix algebra: composition and multiplication of matrices are associative, the identity acts as $I_{n} x = x$ , and $(A B)^{T} = B^{T} A^{T}$ ; the inverse of an invertible square matrix is unique and two-sided (Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, Invertible matrices and the general linear group ${GL}_{n} (F)$ , In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided).

Proof

technique · direct; transfer the positivity statement to operators and use the unique non-negative square root

1.1givenF1F2algebra

By [F1] $M$ corresponds to a positive operator on $R^{n}$ , hence to a non-negative one. By [F2] there is a unique non-negative operator $R$ with $R^{2} = M$ ; write $M^{1 / 2}$ for its matrix, which is symmetric because $R$ is self-adjoint [F1]. If $R v = 0$ then $M v = R^{2} v = 0$ , so $v = 0$ because $M$ is positive definite; hence $R$ is injective, and by [F2] its eigenvalues are non-negative and none is zero, so all are positive: $R$ is positive definite. This proves clause 2 except for uniqueness among all symmetric positive definite square roots, which follows because a symmetric positive definite square root is in particular a non-negative square root, and [F2] gives uniqueness among non-negative square roots.

2.1step 1.1F2F3algebra

In the orthonormal eigenbasis of $R = M^{1 / 2}$ from step 1.1, write $R e_{j} = r_{j} e_{j}$ with $r_{j} > 0$ . Define $B e_{j} = r_{j}^{- 1} e_{j}$ and extend linearly. Then $B R = R B = I_{n}$ on this basis, and $B^{2} M = M B^{2} = I_{n}$ since $M = R^{2}$ . Thus $R$ and $M$ are invertible by [F3]. Since $M = M^{1 / 2} M^{1 / 2}$ [F3], we have $M^{- 1} = (M^{1 / 2})^{- 1} (M^{1 / 2})^{- 1}$ , and $M^{- 1 / 2} : = (M^{1 / 2})^{- 1}$ is well defined and satisfies $(M^{- 1 / 2})^{2} = M^{- 1}$ and $M^{- 1 / 2} M M^{- 1 / 2} = (M^{1 / 2})^{- 1} M^{1 / 2} M^{1 / 2} (M^{1 / 2})^{- 1} = I_{n}$ by associativity and [F3].

2.2step 1.12.1F1F3algebra

$M^{- 1}$ and $M^{- 1 / 2}$ are symmetric and positive definite. For $M^{- 1}$ : for $u, u^{'} \in R^{n}$ write $u = M x$ , $u^{'} = M x^{'}$ ; then $⟨ M^{- 1} u, u^{'} ⟩ = ⟨ x, M x^{'} ⟩ = ⟨ M x, x^{'} ⟩ = ⟨ u, M^{- 1} u^{'} ⟩$ by symmetry of $M$ , so $M^{- 1}$ is self-adjoint, hence symmetric [F1]; and for $u \neq 0$ , $⟨ M^{- 1} u, u ⟩ = ⟨ x, M x ⟩ > 0$ , so $M^{- 1}$ is positive definite. For $M^{- 1 / 2}$ : it is the inverse of the self-adjoint invertible $M^{1 / 2}$ , and the same computation with $M^{1 / 2}$ in place of $M$ (which is positive definite by step 1.1) shows that its inverse is self-adjoint and positive definite. This proves clauses 1 and 3.

3.1step 2.2F1F3algebra∎

Clause 4: for symmetric $K$ , $S^{T} = (M^{- 1 / 2})^{T} K^{T} (M^{- 1 / 2})^{T} = M^{- 1 / 2} K M^{- 1 / 2} = S$ by symmetry of $M^{- 1 / 2}$ and $K$ , so $S$ is symmetric [F3]. For $x \in R^{n}$ put $y : = M^{- 1 / 2} x$ ; then $x^{T} S x = y^{T} K y$ , so non-negativity of $K$ implies non-negativity of $S$ . If $K$ is positive definite and $x \neq 0$ , then $y = M^{- 1 / 2} x \neq 0$ because $M^{- 1 / 2}$ is invertible, and $x^{T} S x = y^{T} K y > 0$ . Thus $S$ is positive definite. Conversely, if $S$ is positive definite and $x \neq 0$ , then with $y : = M^{1 / 2} x \neq 0$ one has $x^{T} K x = y^{T} M^{- 1 / 2} K M^{- 1 / 2} y = y^{T} S y > 0$ , so $K$ is positive definite.

Remarks

The matrices $M^{1 / 2}$ , $M^{- 1 / 2}$ are the standard square roots of a positive definite symmetric matrix; they are used in Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities to reduce a coupled quadratic system to decoupled scalar oscillators, and in Linearisation of the mechanical system at an equilibrium: M(q_0) xi'' + D²U(q_0) xi = 0, and the small-oscillation quadratic model to identify the linearised stiffness matrix.
Clause 4 shows that the eigenvalues of $S$ are non-negative whenever $K$ is non-negative, and positive exactly when $K$ is positive definite. This is what makes the diagonal matrix $Λ$ of Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities have non-negative diagonal entries, with the zero entries corresponding to free (non-oscillating) modes.

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A C² local minimum has vanishing gradient and positive semidefinite Hessian

Statement

Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be open, let $U : Ω \to R$ be of class $C^{2}$ ( $C^{k}$ Euclidean maps and diffeomorphisms), and let $q_{0} \in Ω$ be a local minimum of $U$ : there is $ρ > 0$ with $\overline{B} (q_{0}, ρ) \subseteq Ω$ and $U (q) \geq U (q_{0})$ for every $q \in \overline{B} (q_{0}, ρ)$ . Then

$\nabla U (q_{0}) = 0, h^{T} D^{2} U (q_{0}) h \geq 0 for every h \in R^{n},$

where $D^{2} U (q_{0})$ is the Hessian matrix, which is symmetric (The Hessian of a $C^{2}$ scalar field is symmetric, The Hessian matrix and critical points of a scalar field).

Facts & Assumptions

Given: The $C^{2}$ function $U$ and the local minimum $q_{0}$ .

[F1]

Fermat's theorem in Euclidean space: if a differentiable function has a local extremum at an interior point, its gradient vanishes there (Fermat's theorem: an interior differentiable local extremum has zero gradient, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[F2]

Second-order Taylor expansion: for $C^{2}$ $f$ near $a$ , $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} ⟨ H_{f} (a) h, h ⟩ + o (∥ h ∥^{2})$ , and the Hessian is symmetric (Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$ , The Hessian of a $C^{2}$ scalar field is symmetric, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F3]

The Peano remainder is a limit statement: writing $f (q_{0} + t h) = f (q_{0}) + t \nabla f (q_{0}) \cdot h + \frac{1}{2} t^{2} h^{T} H_{f} (q_{0}) h + r (t h)$ for small $t$ , one has $r (t h) / ∥ t h ∥_{2}^{2} \to 0$ as $t ↓ 0$ ; by the definition of the function limit (The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ ), for every $ε > 0$ there is $δ > 0$ with $∣ r (t h) ∣ \leq ε t^{2} ∥ h ∥_{2}^{2}$ for $0 < t < δ$ (Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$ , The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

Proof

technique · direct; apply Fermat's theorem and expand along a line

1.1givenF1F2algebra

The restriction $φ_{h} (t) : = U (q_{0} + t h)$ is defined and differentiable for $∣ t ∣$ small, with $φ_{h}^{'} (0) = \nabla U (q_{0}) \cdot h$ by the chain rule; it has a local minimum at $t = 0$ because $U$ does. By the one-variable Fermat theorem (or [F1] applied in one dimension) $φ_{h}^{'} (0) \geq 0$ and $φ_{- h}^{'} (0) \geq 0$ ; since $φ_{- h}^{'} (0) = - \nabla U (q_{0}) \cdot h$ , both inequalities give $\nabla U (q_{0}) \cdot h = 0$ . As this holds for every $h$ , $\nabla U (q_{0}) = 0$ .

2.1step 1.1F2F3algebra∎

Fix $h \in R^{n}$ , assume $h \neq 0$ (the case $h = 0$ is trivial), and apply [F2] at $a = q_{0}$ to the vector $t h$ with $t > 0$ small: $U (q_{0} + t h) = U (q_{0}) + t \nabla U (q_{0}) \cdot h + \frac{1}{2} t^{2} h^{T} D^{2} U (q_{0}) h + o (t^{2}) .$ By step 1.1 the linear term vanishes, and the left side is $\geq U (q_{0})$ ; hence, using the notation of [F3], $\frac{1}{2} t^{2} h^{T} D^{2} U (q_{0}) h + r (t h) \geq 0$ for all sufficiently small $t > 0$ . Suppose for contradiction that $h^{T} D^{2} U (q_{0}) h = - c < 0$ . Choose $ε : = c / (4 ∥ h ∥_{2}^{2})$ in [F3]; then for $0 < t < δ$ , $r (t h) \leq ε t^{2} ∥ h ∥_{2}^{2} = c t^{2} / 4$ , so the left side is at most $- \frac{1}{2} c t^{2} + \frac{1}{4} c t^{2} = - \frac{1}{4} c t^{2} < 0$ , a contradiction. Hence $h^{T} D^{2} U (q_{0}) h \geq 0$ .

Remarks

The converse is false without a strictness hypothesis: a $C^{2}$ function with $\nabla U (q_{0}) = 0$ and positive semidefinite Hessian need not have a local minimum at $q_{0}$ . The lemma records only the necessary conditions, which is what the mechanical energy-stability theorem and the linearisation proposition use.
Nothing is asserted about strict positivity of the Hessian: a strict minimum may have a degenerate Hessian (for example $U (q) = ∣ q ∣^{4}$ at $q_{0} = 0$ ), and the energy-stability theorem is stated for strict minima without any Hessian hypothesis.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities

Statement

Let $n \geq 1$ , let $M \in M_{n} (R)$ be symmetric positive definite and let $K \in M_{n} (R)$ be symmetric and non-negative, that is $v^{T} K v \geq 0$ for all $v \in R^{n}$ . Let $M^{1 / 2}$ , $M^{- 1 / 2}$ be the square roots of Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity and let

$S : = M^{- 1 / 2} K M^{- 1 / 2} .$

By the real spectral theorem choose an orthogonal matrix $O$ and a diagonal matrix $Λ = diag (λ_{1}, \dots, λ_{n})$ with $S = O Λ O^{T}$ and $λ_{i} \geq 0$ for all $i$ .

Let $I \subseteq R$ contain at least two points and satisfy $r, s \in I, r \leq u \leq s ⟹ u \in I$ for all $r, s, u \in R$ ; suppose also $0 \in I$ . Let $q : I \to R^{n}$ be twice differentiable with continuous first and second domain-relative derivatives, in the sense of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral. Put $z : = O^{T} M^{1 / 2} q$ . Then:

$q$ satisfies the quadratic system $M q^{''} + K q = 0$ on $I$ if and only if $z$ satisfies the decoupled system

$z_{i}^{''} + λ_{i} z_{i} = 0 (1 \leq i \leq n)$

on $I$ ; 2. for every $i$ the solutions of the scalar equation in clause 1 on an interval containing $0$ , with prescribed initial data, are exactly as follows:

$z_{i} (t) = a_{i} \cos (ω_{i} t) + b_{i} \sin (ω_{i} t) with ω_{i} : = \sqrt{λ_{i}} > 0 when λ_{i} > 0,$

equivalently $z_{i} (t) = Re (C_{i} e^{i ω_{i} t})$ with $C_{i} = a_{i} - i b_{i} \in C$ , and

$z_{i} (t) = a_{i} + b_{i} t when λ_{i} = 0;$

the coefficients are $a_{i} = z_{i} (0)$ , $b_{i} = z_{i}^{'} (0) / ω_{i}$ for $λ_{i} > 0$ and $b_{i} = z_{i}^{'} (0)$ for $λ_{i} = 0$ , so the displayed families cover every initial datum and are the complete solution sets for the first-order system $(z_{i}, z_{i}^{'})^{'} = (z_{i}^{'}, - λ_{i} z_{i})$ ; 3. consequently every solution of $M q^{''} + K q = 0$ is $q = M^{- 1 / 2} O z$ for a unique choice of the modal solutions of clause 2, and the correspondence between initial data $(q (0), q^{'} (0)) \in R^{2 n}$ and the coefficient tuples $(a_{i}, b_{i})$ is a bijection; 4. for every solution the modal energies $E_{i} : = \frac{1}{2} (z_{i}^{' 2} + λ_{i} z_{i}^{2})$ are constant on $I$ , and so are

$\frac{1}{2} {\dot{q}}^{T} M \dot{q} + \frac{1}{2} q^{T} K q = \frac{1}{2} ∣ z^{'} ∣_{2}^{2} + \frac{1}{2} \sum_{i = 1}^{n} λ_{i} z_{i}^{2} = \sum_{i = 1}^{n} E_{i} .$

Facts & Assumptions

Given: The symmetric positive definite $M$ , the symmetric non-negative $K$ , the orthogonal diagonalisation $S = O Λ O^{T}$ with $Λ = diag (λ_{1}, \dots, λ_{n})$ , $λ_{i} \geq 0$ , the set $I$ with the stated interval property, and the curve $q : I \to R^{n}$ with two continuous domain-relative derivatives and $z = O^{T} M^{1 / 2} q$ .

[F1]

$M^{1 / 2}$ , $M^{- 1 / 2}$ exist, are symmetric positive definite, $(M^{1 / 2})^{2} = M$ , $(M^{- 1 / 2})^{2} = M^{- 1}$ and $M^{- 1 / 2} M M^{- 1 / 2} = I_{n}$ ; $S$ is symmetric and non-negative (Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity). A symmetric real matrix is self-adjoint, and the real spectral theorem gives an orthonormal eigenbasis, so $O$ may be taken orthogonal with $O^{T} = O^{- 1}$ and $S = O Λ O^{T}$ (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis, Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

[F2]

Calculus of curves: derivatives are computed componentwise, sums and scalar multiples behave as expected, the chain rule computes derivatives of composites, and a differentiable curve on an interval whose derivative vanishes is constant (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

[F3]

Sine and cosine have derivatives $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ with $\sin 0 = 0$ , $\cos 0 = 1$ ; and $e^{i θ} = \cos θ + i \sin θ$ , so $a \cos θ + b \sin θ = Re ((a - i b) e^{i θ})$ (The derivatives of sine and cosine are cosine and minus sine, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

Every initial value problem of the linear first-order system $(z_{i}, z_{i}^{'})^{'} = (z_{i}^{'}, - λ_{i} z_{i})$ satisfies the Picard-Lindelöf hypotheses, so it has a unique maximal solution; a solution defined on all of $R$ through the given data is therefore the maximal solution (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, First-order systems, initial value problems, and solutions on intervals).

Proof

technique · direct; transform the system by the square root of $M$, diagonalise the transformed stiffness and solve the scalar equations with a uniqueness argument

1.1givenF1F2algebra

Since $q = M^{- 1 / 2} O z$ (equivalently $z = O^{T} M^{1 / 2} q$ ), differentiating twice with the chain rule [F2] gives $q^{''} = M^{- 1 / 2} O z^{''}$ , hence $M q^{''} + K q = M^{1 / 2} O z^{''} + K M^{- 1 / 2} O z = M^{1 / 2} O (z^{''} + O^{T} M^{- 1 / 2} K M^{- 1 / 2} O z) = M^{1 / 2} O (z^{''} + Λ z),$ where we used $M = M^{1 / 2} M^{1 / 2}$ , $O^{T} = O^{- 1}$ and $S = O Λ O^{T}$ [F1]. As $M^{1 / 2} O$ is invertible, $M q^{''} + K q = 0$ if and only if $z^{''} + Λ z = 0$ , which is the decoupled system of clause 1.

1.2F2F3algebra

The scalar equation for $λ_{i} > 0$ : by [F3] and the chain rule, $\frac{d^{2}}{d t^{2}} (a \cos ω_{i} t + b \sin ω_{i} t) = - ω_{i}^{2} (a \cos ω_{i} t + b \sin ω_{i} t) = - λ_{i} z_{i} (t),$ so the displayed family solves $z_{i}^{''} + λ_{i} z_{i} = 0$ on $R$ . For $λ_{i} = 0$ the affine functions solve $z_{i}^{''} = 0$ by direct differentiation. In both cases the coefficients listed in clause 2 realize the initial data $(z_{i} (0), z_{i}^{'} (0))$ .

2.1step 1.2F2F4algebra

Completeness in the scalar case: let $u$ solve $u^{''} + λ_{i} u = 0$ with $u (0) = u^{'} (0) = 0$ . For $λ_{i} > 0$ define $Φ (t) : = \frac{1}{2} u^{'} (t)^{2} + \frac{1}{2} λ_{i} u (t)^{2}$ ; by [F2], $Φ^{'} = u^{'} (u^{''} + λ_{i} u) = 0$ , so $Φ$ is constant and equal to $Φ (0) = 0$ ; both summands are non-negative, so $u = 0$ . For $λ_{i} = 0$ , $u^{''} = 0$ gives $u^{'} = c o n s t = 0$ and then $u = c o n s t = 0$ by [F2]. Hence two solutions with the same initial data coincide on the intersection of their intervals. Now let $z$ be any maximal solution of the scalar first-order system; the explicit solution $w$ of clause 2 with the same initial data is defined on all of $R$ , and by uniqueness $w$ is the restriction of $z$ ; since $w$ is defined on $R$ , the maximal domain is $R$ and $z = w$ . Thus the displayed families are exactly the solution sets, which is clause 2.

2.2step 1.12.1F1algebra

Clause 3 follows from clauses 1 and 2 together with the identity $q = M^{- 1 / 2} O z$ : the map $(q (0), q^{'} (0)) \mapsto (z (0), z^{'} (0))$ is the linear bijection given by $z (0) = O^{T} M^{1 / 2} q (0)$ , $z^{'} (0) = O^{T} M^{1 / 2} q^{'} (0)$ , and the map $(z (0), z^{'} (0)) \mapsto (a_{i}, b_{i})$ is bijective by the formulas of clause 2.

3.1step 1.1F1F2algebra∎

Clause 4: for a solution, $\frac{d}{d t} E_{i} = z_{i}^{'} (z_{i}^{''} + λ_{i} z_{i}) = 0$ by [F2], so each $E_{i}$ is constant. Moreover, using $q^{'} = M^{- 1 / 2} O z^{'}$ and [F1], ${\dot{q}}^{T} M \dot{q} = (M^{- 1 / 2} O z^{'})^{T} M (M^{- 1 / 2} O z^{'}) = ∣ z^{'} ∣_{2}^{2}, q^{T} K q = (M^{- 1 / 2} O z)^{T} K (M^{- 1 / 2} O z) = z^{T} Λ z = \sum_{i} λ_{i} z_{i}^{2},$ because $O^{T} O = I_{n}$ and $M^{- 1 / 2} M M^{- 1 / 2} = I_{n}$ . Adding the two identities and halving gives the displayed chain of equalities.

Remarks

The transformation is the one used for small oscillations: $O$ diagonalises the symmetric matrix $M^{- 1 / 2} K M^{- 1 / 2}$ , whose eigenvalues $λ_{i}$ are non-negative by Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity clause 4. A zero eigenvalue produces a free affine mode $a_{i} + b_{i} t$ , not an oscillation, and negative eigenvalues cannot occur for non-negative $K$ ; for an indefinite $K$ the corresponding scalar equation has real exponential solutions.
Clause 1 is an exact equivalence for the constant-coefficient system. It asserts nothing about a configuration-dependent kinetic metric or a nonlinear potential; the reduction of a small-oscillation linearisation to this theorem is the content of Linearisation of the mechanical system at an equilibrium: M(q_0) xi'' + D²U(q_0) xi = 0, and the small-oscillation quadratic model, and the theorem itself makes no long-time approximation claim about any nonlinear system.

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A strict local minimum of the potential gives a Lyapunov stable equilibrium of the mechanical system, by energy conservation

Statement

Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be open, let $M : Ω \to M_{n} (R)$ be a $C^{2}$ field of symmetric positive definite matrices, let $U : Ω \to R$ be $C^{2}$ , let $q_{0} \in Ω$ with $\nabla U (q_{0}) = 0$ , and suppose $q_{0}$ is a strict local minimum of $U$ : there is $r_{0} > 0$ with $\overline{B} (q_{0}, r_{0}) \subseteq Ω$ and

$U (q) > U (q_{0}) (0 < ∥ q - q_{0} ∥_{2} < r_{0}) .$

Let $G$ be the mechanical vector field of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart on $Ω \times R^{n}$ , with energy $E (q, v) = \frac{1}{2} v^{T} M (q) v + U (q)$ . Then $(q_{0}, 0)$ is a Lyapunov stable equilibrium of $(q^{'}, v^{'}) = G (q, v)$ in the sense of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system: for every $ε > 0$ with $\overline{B} ((q_{0}, 0), ε) \subseteq Ω \times R^{n}$ there is $δ > 0$ such that every maximal solution $(q, v)$ with $∥ (q (0), v (0)) - (q_{0}, 0) ∥_{2} < δ$ is defined for all $t \geq 0$ and satisfies

$∥ (q (t), v (t)) - (q_{0}, 0) ∥_{2} < ε (t \geq 0) .$

No asymptotic stability is asserted.

Facts & Assumptions

Given: The $C^{2}$ positive definite kinetic metric field $M$ , the $C^{2}$ potential $U$ , the strict local minimum $q_{0}$ and the mechanical vector field $G$ .

[F1]

$G$ is $C^{1}$ on $Ω \times R^{n}$ : the formula involves $M$ , its first derivatives, $\nabla U$ and $M^{- 1}$ , with $M^{- 1}$ having entries polynomial in the entries of $M$ divided by the nonvanishing continuous function $\det M$ ; hence $G$ is $C^{1}$ (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart, If $\det (A)$ is a unit, then $A^{- 1} = \det (A)^{- 1} adj (A)$ , If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative). A $C^{1}$ map is locally Lipschitz on compact coordinate subsets (A $C^{1}$ map is locally Lipschitz on compact coordinate subsets, Local Lipschitz continuity in the state variable, locally uniform in time and parameters), so every initial value problem has a unique maximal solution (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, Extensions, maximal solutions, maximal intervals, and global solutions).

[F2]

Energy conservation: for every $C^{2}$ solution $(q, v)$ of the mechanical system, $t \mapsto E (q (t), v (t))$ is constant (Energy is conserved along solutions of the mechanical system with a configuration-dependent kinetic metric).

[F3]

Compact continuation: an autonomous solution trapped in a compact subset of the open state domain near a finite endpoint extends past that endpoint (Compact continuation for locally Lipschitz ODEs: a solution whose graph stays in a compact subset of the domain near a finite endpoint extends past it).

[F4]

Compactness and extrema: closed bounded subsets of $R^{2 n}$ are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line) and a continuous real function on a nonempty compact set attains a minimum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value); the Euclidean norm is a norm, so the triangle inequality and the reverse triangle inequality hold (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation); the reverse inequality follows by applying the triangle inequality in both orders.

Proof

technique · direct; use the energy as a Lyapunov function on the boundary of a ball and extend the trapped solution by compact continuation

1.1givenF4algebra

Fix an admissible test radius $ε_{t e s t} > 0$ and choose $0 < ε \leq ε_{t e s t}$ with $ε < r_{0}$ and $\overline{B} ((q_{0}, 0), ε) \subseteq Ω \times R^{n}$ ; then the projection of the ball to the first factor lies in $\overline{B} (q_{0}, ε) \subseteq \overline{B} (q_{0}, r_{0})$ . Write $E_{0} : = E (q_{0}, 0) = U (q_{0})$ . On the sphere $S_{ε} : = {(q, v) : ∥ (q, v) - (q_{0}, 0) ∥_{2} = ε}$ the continuous function $E - E_{0}$ is strictly positive: if $v \neq 0$ then $\frac{1}{2} v^{T} M (q) v > 0$ , and if $v = 0$ then $0 < ∥ q - q_{0} ∥_{2} = ε < r_{0}$ and $U (q) > U (q_{0})$ . As $S_{ε}$ is nonempty and compact by [F4], there is $δ_{0} : = \min_{S_{ε}} (E - E_{0}) > 0$ .

2.1step 1.1F4algebra

Continuity of $E$ at $(q_{0}, 0)$ gives $δ > 0$ such that $∥ (q, v) - (q_{0}, 0) ∥_{2} < δ$ implies $E (q, v) - E_{0} < δ_{0}$ . Shrinking $δ$ further, assume $δ \leq ε$ .

3.1step 1.12.1F1F2F4algebra

Let $(q, v) : [0, b) \to Ω \times R^{n}$ be the maximal forward solution with $∥ (q (0), v (0)) - (q_{0}, 0) ∥_{2} < δ$ . Then $∥ (q (t), v (t)) - (q_{0}, 0) ∥_{2} < ε$ for all $t \in [0, b)$ . Indeed, suppose not and let $t_{1} : = \inf {t \in (0, b) : ∥ (q (t), v (t)) - (q_{0}, 0) ∥_{2} \geq ε}$ ; since the radius function is continuous with value $< ε$ at $t = 0$ and $\geq ε$ at some time, the intermediate value theorem (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ ) shows that the infimum is attained with $∥ (q (t_{1}), v (t_{1})) - (q_{0}, 0) ∥_{2} = ε$ and $t_{1} > 0$ . By [F2], $E (q (t_{1}), v (t_{1})) = E (q (0), v (0))$ , so $E - E_{0}$ at time $t_{1}$ is $< δ_{0}$ by step 2.1, contradicting the definition of $δ_{0}$ on $S_{ε}$ .

4.1step 3.1F3

The solution is defined for all $t \geq 0$ : if $b < \infty$ , then the values $(q (t), v (t))$ for $t \in [0, b)$ lie in the compact set $\overline{B} ((q_{0}, 0), ε) \subseteq Ω \times R^{n}$ by step 3.1, so [F3] extends the solution past $b$ , contradicting maximality of the maximal solution (Extensions, maximal solutions, maximal intervals, and global solutions).

5.1step 2.13.14.1∎

Therefore every maximal solution starting at distance $< δ$ from $(q_{0}, 0)$ is defined for all $t \geq 0$ and stays at distance $< ε$ ; the resulting bound is also $< ε_{t e s t}$ , and since $ε_{t e s t}$ was arbitrary, $(q_{0}, 0)$ is Lyapunov stable. This is exactly the defining condition of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system: the $δ$ produced in step 2.1 depends only on $ε$ , not on the solution.

Remarks

The proof uses exact energy conservation to establish stability, but it does not prove attraction. In this strict-minimum setting, nearby non-equilibrium initial data have energy greater than $E (q_{0}, 0)$ ; conservation therefore prevents their solutions from converging to $(q_{0}, 0)$ . The theorem makes no asymptotic-stability claim, in contrast with An equilibrium whose linearisation has all eigenvalues in the open left half-plane is locally exponentially stable, which uses a strict Lyapunov decrease.
Neither strict positivity of the Hessian nor a Taylor expansion is used: a degenerate strict minimum such as $U (q) = ∥ q ∥_{2}^{4}$ is covered. The strictness hypothesis on the ball is essential for the positivity of $E - E_{0}$ on the sphere; a non-strict minimum with a flat direction, such as $U \equiv c o n s t$ near $q_{0}$ , would give $E - E_{0} = 0$ at points of the sphere with $v = 0$ and the argument would not apply.

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Linearisation of the mechanical system at an equilibrium: M(q_0) xi'' + D²U(q_0) xi = 0, and the small-oscillation quadratic model

Statement

Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be open, let $M : Ω \to M_{n} (R)$ be a $C^{2}$ field of symmetric positive definite matrices, let $U : Ω \to R$ be $C^{2}$ , and let $q_{0} \in Ω$ be a mechanical equilibrium, that is $\nabla U (q_{0}) = 0$ (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart). Let $G$ be the mechanical vector field on $Ω \times R^{n}$ . Then:

$G$ is totally differentiable at $(q_{0}, 0)$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder) and, writing $D^{2} U (q_{0})$ for the Hessian matrix (The Hessian matrix and critical points of a scalar field),

$D G (q_{0}, 0) (δ q, δ v) = (δ v, - M (q_{0})^{- 1} D^{2} U (q_{0}) δ q) (δ q, δ v \in R^{n});$

consequently the linearised equation at the equilibrium is

$ξ^{''} = - M (q_{0})^{- 1} D^{2} U (q_{0}) ξ, equivalently M (q_{0}) ξ^{''} + D^{2} U (q_{0}) ξ = 0,$

a constant-coefficient system whose mass matrix $M (q_{0})$ is symmetric positive definite and whose stiffness matrix $D^{2} U (q_{0})$ is symmetric (The Hessian of a $C^{2}$ scalar field is symmetric); 3. if in addition $q_{0}$ is a strict local minimum of $U$ , then $K : = D^{2} U (q_{0})$ is non-negative, so the hypotheses of Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities are satisfied by $M (q_{0})$ and $K$ , and the linearised system is exactly the constant-coefficient quadratic system of that theorem with mass matrix $M (q_{0})$ and stiffness $K$ ; its modes are the normal modes of the linearisation.

Facts & Assumptions

Given: The $C^{2}$ kinetic metric field $M$ , the $C^{2}$ potential $U$ , the equilibrium $q_{0}$ , and the vector field $G (q, v) = (v, H (q, v))$ with $H (q, v) = M (q)^{- 1} (\frac{1}{2} \nabla_{q} f (q, v) - (D_{q} M (q) [v]) v - \nabla U (q))$ , $f (q, v) = v^{T} M (q) v$ .

[F1]

Regularity: under the stated hypotheses $G$ is of class $C^{1}$ on $Ω \times R^{n}$ (the inverse $M (q)^{- 1}$ has entries polynomial in the entries of $M$ divided by the nonvanishing continuous function $\det M (q)$ ), hence totally differentiable at every point (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart, If $\det (A)$ is a unit, then $A^{- 1} = \det (A)^{- 1} adj (A)$ , If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, $C^{k}$ Euclidean maps and diffeomorphisms).

[F2]

At $(q_{0}, 0)$ : $\nabla_{q} f (q_{0}, 0) = 0$ and its $q$ -derivative vanishes, because $\partial_{k} (v^{T} M (q) v) = v^{T} \partial_{k} M (q) v$ is a quadratic form in $v$ vanishing identically at $v = 0$ ; similarly $(D_{q} M (q) [v]) v = \sum_{k} v_{k} \partial_{k} M (q) v$ is quadratic in $v$ , so it and both of its first partial derivatives vanish at $v = 0$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

[F3]

At the equilibrium $\nabla U (q_{0}) = 0$ and the total derivative of $\nabla U$ is the Hessian matrix $D^{2} U (q_{0})$ , symmetric (The Hessian matrix and critical points of a scalar field, The Hessian of a $C^{2}$ scalar field is symmetric, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[F4]

At a strict local minimum, $\nabla U (q_{0}) = 0$ and $D^{2} U (q_{0})$ is non-negative (A C² local minimum has vanishing gradient and positive semidefinite Hessian); and the constant-coefficient system $M ξ^{''} + K ξ = 0$ with $M$ symmetric positive definite and $K$ symmetric non-negative is decoupled by Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities.

Proof

technique · direct; differentiate the explicit components of $G$ at the equilibrium and substitute

1.1givenF1algebra

The linearisation of $G$ at $(q_{0}, 0)$ is $D G (q_{0}, 0) (δ q, δ v) = (δ v, D_{q} H (q_{0}, 0) δ q + D_{v} H (q_{0}, 0) δ v)$ . We compute the two partial derivatives of $H = M^{- 1} B$ with $B (q, v) : = \frac{1}{2} \nabla_{q} f (q, v) - (D_{q} M (q) [v]) v - \nabla U (q)$ .

2.1step 1.1F2F3algebra

At $(q_{0}, 0)$ , $B (q_{0}, 0) = \frac{1}{2} \nabla_{q} f (q_{0}, 0) - 0 - \nabla U (q_{0}) = 0$ by [F2] and the equilibrium condition. Hence $D_{q} H (q_{0}, 0) δ q = D_{q} (M^{- 1}) (q_{0}) [δ q] B (q_{0}, 0) + M (q_{0})^{- 1} D_{q} B (q_{0}, 0) δ q = M (q_{0})^{- 1} D_{q} B (q_{0}, 0) δ q,$ and by [F2] and [F3] the derivative of $B$ in $q$ at $(q_{0}, 0)$ is $- \frac{1}{2} \cdot 0 + 0 - D^{2} U (q_{0}) = - D^{2} U (q_{0})$ . Hence $D_{q} H (q_{0}, 0) = - M (q_{0})^{- 1} D^{2} U (q_{0})$ .

2.2step 1.12.1F2algebra

Similarly $D_{v} H (q_{0}, 0) = M (q_{0})^{- 1} D_{v} B (q_{0}, 0)$ because $B (q_{0}, 0) = 0$ again kills the derivative of $M^{- 1}$ along $v$ ; and $D_{v} B (q_{0}, 0) = 0$ because both $\frac{1}{2} \nabla_{q} f$ and $(D_{q} M (q) [v]) v$ are quadratic in $v$ with vanishing first $v$ -derivative at $v = 0$ [F2]. Therefore $D G (q_{0}, 0) (δ q, δ v) = (δ v, - M (q_{0})^{- 1} D^{2} U (q_{0}) δ q)$ , which is clause 1.

2.3step 1.12.2F3algebra

Clause 2: the linearised first-order system is $ξ^{'} = η$ , $η^{'} = - M (q_{0})^{- 1} D^{2} U (q_{0}) ξ$ ; eliminating $η$ and multiplying by the invertible matrix $M (q_{0})$ gives $M (q_{0}) ξ^{''} + D^{2} U (q_{0}) ξ = 0$ . The matrix $D^{2} U (q_{0})$ is symmetric by [F3], and $M (q_{0})$ is symmetric positive definite by hypothesis; the stiffness $K : = D^{2} U (q_{0})$ need not be non-negative, and no normal-mode conclusion is drawn at this point.

3.1step 2.3F4∎

Clause 3: if $q_{0}$ is a strict local minimum, [F4] shows that $K = D^{2} U (q_{0})$ is non-negative; with $M = M (q_{0})$ the hypotheses of the quadratic-mode theorem are exactly satisfied (symmetric positive definite mass and symmetric non-negative stiffness), and its clauses describe the normal modes of the linearisation.

Remarks

The linearisation is a mathematical object: it is the derivative of the model field at the equilibrium, and the replacement of the nonlinear system by its linearisation is an approximation whose error is not controlled here. In particular clause 3 does not assert that the nonlinear system's solutions remain close to the normal modes for all time; the exact quadratic calculation alone gives no long-time error bound.
If the equilibrium is not a minimum, $D^{2} U (q_{0})$ may have positive and negative eigenvalues; the linearised modes then include real exponential directions. Stability conclusions from such a linearisation are the subject of An equilibrium whose linearisation has all eigenvalues in the open left half-plane is locally exponentially stable and A hyperbolic equilibrium whose linearisation has an eigenvalue with positive real part is unstable, which require hyperbolicity and do not apply to the pure oscillation case of clause 3.

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Scope of the quadratic-mode and energy-stability results: exact quadratic model versus linearisation approximation

Statement

The three results of this group establish different, and differently qualified, statements.

Decoupling of the constant-coefficient quadratic system $M q^{''} + K q = 0$ into normal modes, with the zero-mode and energy identities is an exact theorem about the constant-coefficient linear system $M q^{''} + K q = 0$ with $M$ symmetric positive definite and $K$ symmetric non-negative: it diagonalises the system, gives all solutions and the modal energies. It is not an approximation statement, and it makes no claim about nonlinear systems or about configuration-dependent kinetic metrics.
Linearisation of the mechanical system at an equilibrium: M(q_0) xi'' + D²U(q_0) xi = 0, and the small-oscillation quadratic model computes the derivative of the nonlinear mechanical field at an equilibrium. Its linearised equation is the exact first-order Taylor model at that point; replacing the nonlinear system by it is an approximation whose error over a time interval is not estimated by the proposition. A zero eigenvalue $λ_{i} = 0$ of $M (q_{0})^{- 1 / 2} D^{2} U (q_{0}) M (q_{0})^{- 1 / 2}$ corresponds to an affine mode of the linearisation, not to an oscillation, and a negative eigenvalue corresponds to exponential growth of the linearisation.
A strict local minimum of the potential gives a Lyapunov stable equilibrium of the mechanical system, by energy conservation is an exact Lyapunov-stability theorem for the nonlinear system at a strict minimum of the potential. It is a statement about the nonlinear field of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart, uses exact energy conservation, and therefore proves stability but not attraction: the theorem does not assert that solutions converge to the equilibrium, and in a conservative model with no dissipation they do not in general. It also does not assert that the linearisation determines the nonlinear stability in non-hyperbolic cases, and it does not require the Hessian to be positive definite.

5 · Examples, experiments and counterexamples

None yet.