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Hyperbolic linearisation of an equilibrium of the model: exponential dichotomy of the linearised flow, local exponential asymptotic stability in the stable case, and instability in the saddle case

Physical scope

The consequences of the hyperbolic linearisation of an equilibrium state of an autonomous model whose force map is of class C1 in the state: invariant stable and unstable subspaces of the linearised flow with exponential estimates and exponential separation from the equilibrium; local exponential asymptotic stability of the equilibrium when the whole spectrum lies in the open left half-plane; and instability when no eigenvalue lies on the imaginary axis and one lies in the open right half-plane. The rates and constants belong to the linearised flow in the declared state chart; no centre-case, global, parameter-uniform or empirical statement is made.

Statement

Let an autonomous stability model be fixed whose force map is of class $C^{1}$ in the state, with state domain $S_{0} \subseteq R^{6 N}$ , associated $C^{1}$ field $G$ and local flow $Φ : Ω \to S_{0}$ (The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets clauses (i)-(ii)); all differentiability is that of $C^{k}$ Euclidean maps and diffeomorphisms. Let $z_{*} \in S_{0}$ be an equilibrium state, $G (z_{*}) = 0$ , and let $A : = D G (z_{*})$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder). All state and tangent norms below are Euclidean norms of the dimensionless SI numerical coordinates $z = (X / (1 m), V / (1 m s^{- 1}))$ fixed by The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets clause (iv), and $t = t_{p h y s} / t_{r e f}$ with $t_{r e f} = 1 s$ . Thus $G$ and $A$ are the numerical field and its derivative in that chart; no norm adds physical positions to physical velocities. Then:

(Exponential dichotomy.) If 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 $0$ --- that is, if $z_{*}$ is hyperbolic in the sense of Stability vocabulary of the model: Lyapunov, asymptotic and orbital stability, linear stability of equilibria and map fixed points, and hyperbolicity of equilibria and fixed points clause (iv) --- then there are $A$ -invariant subspaces $E_{s}, E_{u} \subseteq R^{6 N}$ with $R^{6 N} = E_{s} \oplus E_{u}$ and real constants $C \geq 1$ , $c > 0$ such that, with the Euclidean norm of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and the exponential series $e^{t A}$ : $∥ e^{t A} v ∥_{2} \leq C e^{- c t} ∥ v ∥_{2} (v \in E_{s}, t \geq 0), ∥ e^{t A} v ∥_{2} \geq C^{- 1} e^{c t} ∥ v ∥_{2} (v \in E_{u}, t \geq 0),$ $∥ e^{t A} v ∥_{2} \leq C e^{c t} ∥ v ∥_{2} (v \in E_{u}, t \leq 0),$ and $E_{s} = {v : e^{t A} v \to 0 (t \to + \infty)}$ , $E_{u} = {v : e^{t A} v \to 0 (t \to - \infty)}$ . The spaces $E_{s}, E_{u}$ are the stable and unstable subspaces of the linearised flow.
(Stable case.) If every complex eigenvalue of $A$ has negative real part, then $z_{*}$ is locally exponentially asymptotically stable: there are $δ > 0$ , $β \geq 1$ , $γ > 0$ such that every maximal solution $z$ of the model with $∥ z (0) - z_{*} ∥_{2} < δ$ is defined for all $t \geq 0$ and satisfies $∥ z (t) - z_{*} ∥_{2} \leq β e^{- γ t} ∥ z (0) - z_{*} ∥_{2} (t \geq 0) .$ In particular $z_{*}$ is Lyapunov stable and asymptotically stable (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system).
(Saddle case.) If no complex eigenvalue of $A$ has real part $0$ and at least one has positive real part, then $z_{*}$ is unstable: there is $ε_{0} > 0$ such that for every $δ > 0$ there are a state $z_{0}$ with $∥ z_{0} - z_{*} ∥_{2} < δ$ and a time $t_{1} > 0$ such that the maximal solution through $z_{0}$ is defined on $[0, t_{1}]$ and $∥ z (t_{1}) - z_{*} ∥_{2} = ε_{0}$ .
(Rates, charts and scope.) The constants $C, c, β, γ$ and the coordinate descriptions of the subspaces depend on the chosen chart and norm; the physical rates $c / t_{r e f}, γ / t_{r e f}$ have dimension $τ^{- 1}$ . Under a smooth change of chart the derivative matrices at the equilibrium are similar, so their eigenvalue spectrum is invariant, while the Euclidean estimates may change as recorded in Stability vocabulary of the model: Lyapunov, asymptotic and orbital stability, linear stability of equilibria and map fixed points, and hyperbolicity of equilibria and fixed points clause (vi)(b). No conclusion is asserted when some eigenvalue has zero real part (the centre case), and no statement is made about the size of the stable set, the basin of attraction, uniformity in a family of models or any empirical realization.

Facts & Assumptions

Given: the autonomous model, its $C^{1}$ field $G$ , the equilibrium $z_{*}$ and $A = D G (z_{*})$ ; for clause 1 the hyperbolicity hypothesis, for clause 2 the negative-spectrum hypothesis, for clause 3 the saddle hypothesis.

[L1]

The linear dichotomy (Exponential dichotomy for a hyperbolic endomorphism: invariant stable and unstable subspaces with exponential estimates): for an endomorphism $T$ of a finite-dimensional real inner product space with no eigenvalue of its complexification of real part $0$ , there are $T$ -invariant subspaces $E_{s}, E_{u}$ with $V = E_{s} \oplus E_{u}$ and constants $C \geq 1$ , $c > 0$ satisfying exactly the four estimates and the two limit characterisations displayed in clause 1 of the Statement.

[L2]

Exponential stability in the stable case (An equilibrium whose linearisation has all eigenvalues in the open left half-plane is locally exponentially stable): if $F : U \to R^{n}$ is $C^{1}$ on an open $U ∋ 0$ with $F (0) = 0$ and every eigenvalue of $D F (0)$ has negative real part, then there are $δ > 0$ , $β \geq 1$ , $γ > 0$ such that every maximal solution with $∥ z (0) ∥_{2} < δ$ is defined for all $t \geq 0$ and satisfies $∥ z (t) ∥_{2} \leq β e^{- γ t} ∥ z (0) ∥_{2}$ .

[L3]

Instability in the saddle case (A hyperbolic equilibrium whose linearisation has an eigenvalue with positive real part is unstable): if $F : U \to R^{n}$ is $C^{1}$ on an open $U ∋ 0$ with $F (0) = 0$ , no eigenvalue of $D F (0)$ has real part $0$ and at least one has positive real part, then there is $ε_{0} > 0$ such that for every $δ > 0$ there are $z_{0}$ with $∥ z_{0} ∥_{2} < δ$ and $t_{1} > 0$ with the maximal solution of $z^{'} = F (z)$ through $z_{0}$ defined on $[0, t_{1}]$ and $∥ z (t_{1}) ∥_{2} = ε_{0}$ .

Proof

technique · direct

1.1givenL1

Clause 1: the field $G$ is $C^{1}$ in the state by hypothesis, so $A = D G (z_{*})$ is an endomorphism of $V = R^{6 N}$ with its Euclidean inner product; the hyperbolicity hypothesis is exactly the hypothesis of [L1] on the complexified eigenvalues. Applying [L1] with $T : = A$ gives the two invariant subspaces, the four exponential estimates and the two limit characterisations of clause 1.

1.2givenalgebra

Translation to the origin: define $U : = S_{0} - z_{*} = {ζ : ζ + z_{*} \in S_{0}}$ , an open set containing $0$ , and $F : U \to R^{6 N}$ , $F (ζ) : = G (ζ + z_{*})$ . Then $F$ is $C^{1}$ , $F (0) = G (z_{*}) = 0$ , and $D F (0) = D G (z_{*}) = A$ .

2.1step 1.2givenL2algebra

Clause 2: under the negative-spectrum hypothesis, [L2] applied to the field $F$ of step 1.2 gives $δ, β, γ$ with $∥ ζ (t) ∥_{2} \leq β e^{- γ t} ∥ ζ (0) ∥_{2}$ for every maximal solution $ζ$ of $ζ^{'} = F (ζ)$ with $∥ ζ (0) ∥_{2} < δ$ . The curves $t \mapsto ζ (t)$ of $ζ^{'} = F (ζ)$ are exactly the translates $z (t) - z_{*}$ of the model's maximal solutions through states near $z_{*}$ , since $z^{'} = G (z)$ iff $(z - z_{*})^{'} = G ((z - z_{*}) + z_{*})$ ; translating back gives the displayed bound and the all- $t \geq 0$ domain of clause 2, and $[L 2]$ 's conclusion identifies it as Lyapunov and asymptotic stability in the sense of the model's stability vocabulary.

2.2step 1.2givenL3algebra

Clause 3: under the saddle hypothesis, [L3] applied to the field $F$ of step 1.2 yields $ε_{0} > 0$ such that for every $δ > 0$ there are $ζ_{0}$ with $∥ ζ_{0} ∥_{2} < δ$ and $t_{1} > 0$ with $∥ ζ (t_{1}) ∥_{2} = ε_{0}$ ; setting $z_{0} : = ζ_{0} + z_{*}$ gives a model datum within distance $δ$ of $z_{*}$ whose maximal solution leaves the ball of radius $ε_{0}$ at time $t_{1}$ . Hence for every ball around $z_{*}$ there is a datum whose solution is not confined to a fixed ball of radius $ε_{0}$ , so $z_{*}$ is not Lyapunov stable, that is unstable.

3.1step 1.1step 2.1step 2.2givenL1L2L3∎

Clause 4: the constants and subspaces produced in steps 1.1, 2.1 and 2.2 are those of [L1], [L2] and [L3] applied to the chart field $G$ ; a $C^{2}$ chart transition conjugates the derivative at $z_{*}$ by its Jacobian because $G (z_{*}) = 0$ , so the derivative matrices are similar and have the same spectrum (The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets clause (vi)(b)). The Euclidean estimates may change with the chart, but all three supplier applications are valid for the same hyperbolicity and spectral hypotheses. The hypotheses of [L1], [L2] and [L3] all exclude eigenvalues on the imaginary axis, and no step above produced a conclusion in that case; no step produced a quantitative basin, a parameter-uniform statement or an empirical assertion.

Remarks

Reading the dichotomy physically. The subspace $E_{s}$ consists of the tangent directions whose linearised deviations from the equilibrium decay exponentially, and $E_{u}$ of those that grow exponentially; the dichotomy quantifies the separation of nearby initial states only at the linear level. For the nonlinear model, clause 2 upgrades the stable case to local exponential attraction and clause 3 upgrades the saddle case to genuine nonlinear instability.
Why the centre case is excluded. If some eigenvalue lies on the imaginary axis, the quadratic Lyapunov form of [L2] is not negative definite and the cone argument of [L3] does not apply; the page records this gap in the scope clause and in Scope of the stability, perturbation and chaos statements: conditional model deductions, the limits of finite-time and numerical evidence, and the empirical qualifications.
Units. $∥ A ∥$ is the induced operator norm on the numerical state space; $∥ A ∥ / t_{r e f}$ and the physical rates $γ / t_{r e f}, c / t_{r e f}$ have dimension $τ^{- 1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). All exponential arguments and all state norms in the estimates are dimensionless. Fixed reference scales affect the numerical bounds, while preserving the qualitative stability predicates.

Hyperbolic linearisation of an equilibrium of the model: exponential dichotomy of the linearised flow, local exponential asymptotic stability in the stable case, and instability in the saddle case

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources