Stability, perturbations and chaos

This item fixes the vocabulary in which the stability, perturbation and chaos statements of this page are phrased. All objects below are model data; nothing here is a measured value, and no clause asserts that a particular model has an equilibrium, a periodic orbit or an invariant set.

(i) The model class. A stability model is a force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N\ge 1$ labels, fixed masses $m_1,\dots ,m_N>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)), a time window $I_0\subseteq \mathbb{R}$ that is a nondegenerate open interval and an open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator). Its full force map $F_0$ has domain $D_0:=I_0\times Q\times {\mathbb{R}}^{3N}$, exactly as in clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance. Put $\mathsf{M}:=\mathrm{diag}(m_1{I}_3,\dots ,m_N{I}_3)$ and define the full first-order field $Z_0:D_0\to {\mathbb{R}}^{6N},Z_0(t,x,v):=(v,{\mathsf{M}}^{-1}{F}_0(t,x,v))\mathrm{.}$ The page may specify an open restriction domain $D\subseteq D_0$ and then uses $F:=F_0{\mathrm{\mid }}_D$; every force evaluation in the statements below is on this restriction. This is still a restriction of the supplier's full force map, not a force map with a different domain. The associated field on this restriction is $Z:=Z_0{\mathrm{\mid }}_D$. The motions are the $C^2$ curves $x:J\to Q$, where $J\subseteq I_0$ is a time interval, of The associated first-order system, motions, maximal motions and the Newtonian flow of a force model for which $(t,x(t),x^{\mathrm{\prime }}(t))\in D$ and $m_i{x}_i^{\mathrm{\prime }\mathrm{\prime }}(t)=F_i(t,x(t),x^{\mathrm{\prime }}(t))$. For any declared open domain $D\subseteq D_0$, including $D=D_0$, call such a motion $D$-maximal when it has no extension to a strictly larger interval whose time-state graph $(t,x(t),x^{\mathrm{\prime }}(t))$ remains in $D$; this is the maximal-solution notion for the first-order ODE $z^{\mathrm{\prime }}=Z(t,z)$ on the open domain $D$ (First-order systems, initial value problems, and solutions on intervals, Extensions, maximal solutions, maximal intervals, and global solutions). A $D$-maximal motion is a restriction of a supplier motion of the full force model, but its maximal interval need not be maximal for the full domain $D_0$. The model is autonomous when $I_0=\mathbb{R}$, $D=\mathbb{R}\times S_0$ for an open state domain $S_0\subseteq Q\times {\mathbb{R}}^{3N}$, and $F$ is independent of $t$. It is $P$-periodic, for $P>0$, when $I_0=\mathbb{R}$, $D$ is invariant under $(t,x,v)\mapsto (t+P,x,v)$ in both directions, and $F(t+P,x,v)=F(t,x,v)$ on $D$. Thus every integer shift by $P$ preserves the time and state domain used by the model. All time intervals use the frame-time parameter fixed by Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors with $[t]=\tau$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

For data $(t_0,x_0,v_0)\in D$, put $z_0:=(x_0,v_0)$. The full field $Z_0$ is continuous and locally Lipschitz in the state, so Every Picard–Lindelöf initial value problem has one maximal solution on an open interval gives a unique maximal solution $z_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}:I_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}\to Q\times {\mathbb{R}}^{3N}$ through $(t_0,z_0)$, where $I_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}$ is an open interval. Let $I_D$ be the connected component containing $t_0$ of $\{t\in I_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}:(t,z_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(t))\in D\}\mathrm{.}$ Because $D$ is open, the preimage is open relative to the open interval $I_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}$, and hence open in $\mathbb{R}$; thus $I_D$ is an open interval. The restriction $z_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}{\mathrm{\mid }}_{I_D}$ is the unique $D$-maximal motion through the data: every motion in $D$ is a solution of the full model and therefore agrees with $z_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}$ on its interval, so no $D$-extension can pass beyond this component.

(ii) Associated field and flow. Write $n:=3N$; the state curve $z:=(x,x^{\mathrm{\prime }})$ of a motion is a solution of $z^{\mathrm{\prime }}=Z(t,z)$ and conversely (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model clause (c)). For an autonomous model write $G:S_0\to {\mathbb{R}}^{6N}$, $G(x,v):=(v,{\mathsf{M}}^{-1}F(x,v))$; by the standing regularity of $F$, the field $G$ is continuous and locally Lipschitz in the state, so it has a local flow $\mathrm{\Phi }:\mathrm{\Omega }⟶S_0,\mathrm{\Phi }(t,z_0):=z(t;z_0),$ with maximal solutions $z(\cdot ;z_0):I_{z_0}\to S_0$ and flow domain $\mathrm{\Omega }=\{(t,z_0):t\in I_{z_0}\}$, in the sense of The local flow of an autonomous first-order system. The state curves of the model's motions through $z_0$ at time $0$ are exactly the restrictions $\mathrm{\Phi }(\cdot ,z_0){\mathrm{\mid }}_J$ to intervals $J\subseteq I_{z_0}$ containing $0$ with nonempty interior, with endpoint derivatives read as in The associated first-order system, motions, maximal motions and the Newtonian flow of a force model clause (c). The unique $D$-maximal motion has state curve $\mathrm{\Phi }(\cdot ,z_0)$ on all of $I_{z_0}$, and $\mathrm{\Phi }(s+t,z_0)=\mathrm{\Phi }(t,\mathrm{\Phi }(s,z_0))$ wherever both sides are defined (The local flow is a partial one-parameter group: identity, composition and inverse-time laws). If in addition $F$ is of class $C^1$ in the state, then $\mathrm{\Omega }$ is open and $\mathrm{\Phi }$ is $C^1$ on it (The flow domain of a C¹ autonomous field is open and the flow is C¹ on it). No completeness of $\mathrm{\Phi }$ is asserted: the maximal interval $I_{z_0}$ may be bounded (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model).

(iii) Equilibria and periodic orbits. A state $z_{\ast }=(x_{\ast },v_{\ast })\in S_0$ is an equilibrium state of the autonomous model when $G(z_{\ast })=0$, that is $v_{\ast }=0$ and $F(x_{\ast },0)=0$; under that condition the constant curve $x(t)=x_{\ast }$ is a motion. A periodic motion of the model is a nonconstant motion $x:\mathbb{R}\to Q$ of a model with $I_0=\mathbb{R}$ satisfying $x(t+T)=x(t)$ for every $t\in \mathbb{R}$ and some $T>0$; the least such positive $T$ is its least period. Its state trace is ${\mathrm{\Gamma }}_z:=\{(x(t),x^{\mathrm{\prime }}(t)):t\in \mathbb{R}\}\subseteq Q\times {\mathbb{R}}^{3N}$. For an autonomous model this trace lies in $S_0$ and is called its state orbit; a periodic orbit or closed orbit means this state orbit. In a nonautonomous periodic model the state trace is not an orbit of a time-independent flow, and the discrete-time objects used on this page are its stroboscopic maps. A limit cycle is an isolated periodic orbit of an autonomous model: there is an open set $U\subseteq S_0$ with ${\mathrm{\Gamma }}_z\subseteq U$ such that every periodic orbit ${\mathrm{\Gamma }}_{\tilde{z}}$ of this model satisfying ${\mathrm{\Gamma }}_{\tilde{z}}\subseteq U$ equals ${\mathrm{\Gamma }}_z$. Thus $U$ contains no other periodic orbit as a whole; an orbit that merely passes through $U$ is not excluded. A limit cycle's attraction is a separate property and is not part of this definition.

(iv) Invariant sets. A set $S\subseteq S_0$ is invariant for the autonomous model when $(t,z)\in \mathrm{\Omega }$ for every $z\in S$ and every $t\in \mathbb{R}$, and ${\mathrm{\Phi }}_t(S)=S$ for all $t\in \mathbb{R}$, where ${\mathrm{\Phi }}_t(z):=\mathrm{\Phi }(t,z)$; it is forward invariant when $(t,z)\in \mathrm{\Omega }$ for every $z\in S$, $t\ge 0$, and ${\mathrm{\Phi }}_t(S)\subseteq S$ for every $t\ge 0$. The orbital distance from a state $z$ to a nonempty set $S$ is $d(z,S):={\inf }_{s\in S}\parallel z-s{\parallel }_2,$ with the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ on the state chart. Precisely, the state coordinates here are the numerical values of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework clause (c): if $X,V$ are the physical position and velocity tuples, the numerical chart is $z=(X\mathrm{/}(1\mathrm{m}),V\mathrm{/}(1\mathrm{m}{\mathrm{s}}^{-1}))$. The norm and orbital distance sum squares of these dimensionless numerical components, rather than adding physical lengths to physical velocities. Time $t$ is the numerical value of physical time in $1\mathrm{s}$. This convention leaves the supplier's numerical ODE unchanged; it fixes the scales for distances.

(v) Homoclinic and heteroclinic orbits. In an autonomous model, a nonconstant complete motion $x:\mathbb{R}\to Q$ with state curve $z:\mathbb{R}\to S_0$ is homoclinic to an equilibrium state $z_{\ast }$ when $z(t)\to z_{\ast }$ as $t\to +\mathrm{\infty }$ and as $t\to -\mathrm{\infty }$, and is heteroclinic from $z_{-}$ to $z_{+}$ for distinct equilibrium states $z_{-},z_{+}$ when $z(t)\to z_{-}$ as $t\to -\mathrm{\infty }$ and $z(t)\to z_{+}$ as $t\to +\mathrm{\infty }$. Both are limit relations of complete model motions, not intersections of separately constructed geometric objects; the differentiability of the associated stable and unstable sets is not asserted here.

(vi) Reading conventions. (a) The vocabulary is attached to the declared model class; equilibria, periodic orbits, invariant sets and homoclinic orbits in clauses (iii)-(v) are objects of autonomous models. A periodic motion of a time-dependent model has a state trace but is not an orbit of an autonomous flow; a periodic model's discrete-time map is defined separately on this page. No clause asserts that a physical system is exactly a stability model or realises any of these objects. (b) The state space is written in a Cartesian chart. Let $h:S_0\to {\tilde{S}}_0$ be a $C^2$ diffeomorphism between open state domains, with time unchanged. If it changes an autonomous field from $G$ to $\tilde{G}(h(z))=Dh(z)G(z)$, and $G$ is $C^1$ near an equilibrium $z_{\ast }$, then its derivative there satisfies $D\tilde{G}(h(z_{\ast }))=Dh(z_{\ast })DG(z_{\ast })Dh(z_{\ast }{)}^{-1}$: the term differentiating $Dh$ vanishes because $G(z_{\ast })=0$. Thus the linearization spectrum at an equilibrium is chart-invariant. This spectrum assertion is conditional on differentiability; the standing locally Lipschitz hypothesis alone does not supply a linearization. Norms, matrix entries and quantitative growth bounds can still depend on the chart. State curves correspond by $z(t)\mapsto h(z(t))$ on the same maximal time intervals: the chain rule gives the transformed equation, and any extension pulls back by $h^{-1}$. Thus equilibria, periodic state orbits and invariant sets correspond, and convergence to a specified state is preserved by continuity of $h$ and $h^{-1}$, including the equilibrium limits in clause (v). Convergence of Euclidean distance to an arbitrary nonempty invariant set requires additional metric control. For example, if constants $0<c\le C<\mathrm{\infty }$ satisfy $c\parallel z-s{\parallel }_2\le \parallel h(z)-h(s){\parallel }_2\le C\parallel z-s{\parallel }_2$ for every $s\in S$ and every state $z$ under consideration, taking infima gives $cd(z,S)\le d(h(z),h(S))\le Cd(z,S)$ and preserves distance-to-set convergence. A smooth diffeomorphism alone supplies no such uniform bounds: for the complete free model $G(x,v)=(v,0)$ on ${\mathbb{R}}^6$, the invariant set $S=\{(x,0):x\in {\mathbb{R}}^3\}$ and motion $z(t)=(t{e}_1,e_1)$, where $e_1=(1,0,0)$, satisfy $d(z(t),S)=1$. The smooth diffeomorphism $h(x,v)=(x,e^{-x_1}v)$, with inverse $(x,w)\mapsto (x,e^{x_1}w)$, instead gives $d(h(z(t)),h(S))=e^{-t}\to 0$. Thus distance-to-set attraction along an escaping motion is not asserted to be chart-invariant.

• Stability vocabulary. Whether an equilibrium state is stable, attracting or unstable is not defined here; those predicates are fixed later on the page, and their statements quantify over the maximal solutions defined in clause (ii).

• Discrete time. For the $P$-periodic model the theory of this page uses a stroboscopic map at the period; that map and the local return maps of transversal sections are fixed in the page's discrete-time definition, and they are not repeated here. Nothing in this definition asserts that such a map is defined on all states or that its iterates exist for all time.

This item attaches stability predicates to two explicitly stated model classes. (a) Framework force models: an autonomous model of 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), with open state domain $S_0\subseteq {\mathbb{R}}^{6N}$, associated field $G:S_0\to {\mathbb{R}}^{6N}$ and its maximal-solution local flow $\mathrm{\Phi }:\mathrm{\Omega }\to S_0$ (The local flow of an autonomous first-order system). (b) Mechanical chart models: the data $(\mathrm{\Omega },M,U)$ of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart, where $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ is open, $M$ is a $C^2$ positive-definite matrix field, and $U$ is $C^2$; here the state domain is $S_0:=\mathrm{\Omega }\times {\mathbb{R}}^n$, the state dimension is $m:=2n$, and the associated mechanical field is $G:S_0\to {\mathbb{R}}^m$. It is $C^1$, hence continuous and locally Lipschitz, and has the local flow and maximal solutions of The local flow of an autonomous first-order system. For class (a) only the standing continuous, state-locally-Lipschitz regularity is required for clauses (i)-(ii); clauses (iii)-(iv) add $C^1$ regularity explicitly. In each application the predicates concern the declared model objects and quantify over its maximal solutions; the two model classes are not being identified.

Numerical state metric. All norms and balls below apply to dimensionless numerical coordinates. For class (a) these are the SI numerical coordinates of 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). For class (b), fix a positive duration reference $t_{\mathrm{r}\mathrm{e}\mathrm{f}}$ and a positive reference quantity $a_j$ of the dimension of each configuration coordinate, and read $q_j=Q_j\mathrm{/}{a}_j$, $v_j=V_j\mathrm{/}(a_j\mathrm{/}{t}_{\mathrm{r}\mathrm{e}\mathrm{f}})$ and $t=t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{t}_{\mathrm{r}\mathrm{e}\mathrm{f}}$; thus $q^{\mathrm{\prime }}=v$. The given mathematical chart data $M,U,G$ are the numerical data in this chart. For either class the distance is $d(z,z^{\mathrm{\prime }})=({\sum }_j(q_j-q_j^{\mathrm{\prime }}{)}^2+{\sum }_j(v_j-v_j^{\mathrm{\prime }}{)}^2{)}^{1\mathrm{/}2}$, with $q=x$ in class (a). The same convention applies to tangent vectors and to clause (v)'s numerical map charts. Different positive reference scales give equivalent norms: their ratios are bounded above and below by the largest and smallest diagonal scale ratios. Hence they preserve the stability, convergence and boundedness predicates, while quantitative radii and bounds can change.

(i) Lyapunov and asymptotic stability. Let $z_{\ast }\in S_0$ be an equilibrium state. The state $z_{\ast }$ is Lyapunov stable when for every Euclidean ball $B_{\epsilon }(z_{\ast })$ with $\overline{B_{\epsilon }(z_{\ast })}\subseteq S_0$ (Open ball, closed ball and sphere in a metric space) there is $\delta \in (0,\epsilon ]$ such that every maximal solution $z$ with $\parallel z(0)-z_{\ast }{\parallel }_2<\delta$ is defined for all $t\ge 0$ and satisfies $\parallel z(t)-z_{\ast }{\parallel }_2<\epsilon$ for all $t\ge 0$. It is asymptotically stable when it is Lyapunov stable and there is ${\delta }_0>0$ such that every maximal solution with $\parallel z(0)-z_{\ast }{\parallel }_2<{\delta }_0$ is defined for all $t\ge 0$ and converges to $z_{\ast }$ as $t\to \mathrm{\infty }$. This is the definition of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system clauses 1-2, applied to the autonomous field $G$ on $S_0$; it is recorded here to fix the exact quantifier order and the requirement that the solutions are defined for all $t\ge 0$. A state that is not Lyapunov stable is unstable.

(ii) Orbital stability of an invariant set. Let $S\subseteq S_0$ be a nonempty compact set invariant under the selected model's local flow $\mathrm{\Phi }$; that is, $(t,z)\in \mathrm{\Omega }$ for every $z\in S$ and $t\in \mathbb{R}$, and ${\mathrm{\Phi }}_t(S)=S$ for every $t\in \mathbb{R}$. For the framework force-model class this is clause (iv) of The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets. The Euclidean orbital distance is $d(z,S):={\inf }_{s\in S}\parallel z-s{\parallel }_2$. Then $S$ is orbitally stable when for every open set $U\supseteq S$ there is an open set $V$ with $S\subseteq V\subseteq U$ such that every maximal solution $z$ with $z(0)\in V$ is defined for all $t\ge 0$ and satisfies $z(t)\in U$ for all $t\ge 0$. Orbital stability is a statement about the set, not about one motion, and it does not assert attraction.

(iii) Linear stability of an equilibrium. Suppose the field $G$ of the chosen model is of class $C^1$ on its state domain $S_0$, so that its local flow is $C^1$ and $DG(z_{\ast })$ exists. Write $m:=\dim S_0$. For each initial deviation $u\in {\mathbb{R}}^m$, let ${\eta }_u:[0,\mathrm{\infty })\to {\mathbb{R}}^m$ be the unique solution of the associated constant-coefficient vector equation ${\eta }_u^{\mathrm{\prime }}(t)=DG(z_{\ast }){\eta }_u(t),{\eta }_u(0)=u$ For the constant reference motion $z(t)=z_{\ast }$, The variational equation along an ODE solution instead defines the normalized matrix equation $A^{\mathrm{\prime }}(t)=DG(z_{\ast })A(t)$, $A(0)=I_m$. Its solution acts on initial deviations by ${\eta }_u(t)=A(t)u$: differentiating this product gives the displayed vector equation and its initial value. The equilibrium $z_{\ast }$ is linearly stable when ${\sup }_{t\ge 0}\parallel {\eta }_u(t){\parallel }_2<\mathrm{\infty }$ for every $u$; it is linearly unstable if this norm is unbounded as $t\to \mathrm{\infty }$ for some $u$. This is a statement about the linear variational equation and is distinct from the nonlinear Lyapunov and asymptotic predicates in clause (i).

(iv) Hyperbolicity of an equilibrium. Let $z_{\ast }$ be an equilibrium state of the chosen autonomous model whose associated field $G$ is $C^1$ on $S_0$, and let $DG(z_{\ast })$ be its derivative. Then $z_{\ast }$ is hyperbolic when no complex eigenvalue of $DG(z_{\ast })$ (Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism) has real part $0$; the eigenvalues are those of the matrix of $DG(z_{\ast })$ in the state chart. The harmonic oscillator equilibrium of a one-dimensional model with a positive stiffness has the purely imaginary eigenvalues $\pm i\omega$ and is therefore not hyperbolic; the saddle of a repulsive linear model has the real eigenvalues $\pm \kappa$, $\kappa >0$, and is hyperbolic.

(v) Linear stability and hyperbolicity of a map. Let $X,Y\subseteq {\mathbb{R}}^m$ be open and let $\mathrm{\Psi }:X\to Y$ be of class $C^1$ ($C^k$ Euclidean maps and diffeomorphisms). A fixed point of this local map is a point $a\in X$ with $\mathrm{\Psi }(a)=a\in Y$. The fixed point is linearly stable when ${\sup }_{n\ge 0}\parallel D\mathrm{\Psi }(a{)}^{n}u{\parallel }_2<\mathrm{\infty }$ for every $u\in {\mathbb{R}}^m$, and hyperbolic when no complex eigenvalue of $D\mathrm{\Psi }(a)$ has modulus $1$. These predicates require only the local map near the fixed point; they concern its derivative, not its nonlinear iterates on everywhere-defined state domains.

(vi) Distinctions. (a) The predicates in clauses (i)-(v) have different definitions and are not interchangeable. In particular, Lyapunov stability does not by definition include attraction, and orbital stability is a property of a set rather than of one of its points. Hyperbolicity alone does not imply linear instability: if all eigenvalues of $DG(z_{\ast })$ have negative real part, the stable estimate of Exponential dichotomy for a hyperbolic endomorphism: invariant stable and unstable subspaces with exponential estimates makes every solution of the variational equation decay and hence remain bounded. If a hyperbolic equilibrium has an eigenvalue with positive real part, the same dichotomy supplies an unbounded forward solution and the equilibrium is linearly unstable. (b) At an equilibrium the spectrum of $DG(z_{\ast })$ is invariant under a $C^2$ chart transition, by the similarity formula in 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); norms and finite-time growth bounds can depend on the chart. (c) For the mechanical chart data $(M,U)$ of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart, at a mechanical equilibrium $(q_0,0)$ the derivative of the field is $DG(q_0,0)(\delta q,\delta v)=(\delta v,-M(q_0{)}^{-1}{D}^{2}U(q_0)\delta q)$; the stability consequences of that computation are proved later on the page.

• Where the map predicates are used. Clause (v) applies to the model's $C^1$ stroboscopic branch and to the local return-map branch expressed in the section charts supplied by the return-map result. Each is a local map between open chart neighborhoods near its fixed point. The predicates concern the derivative there, not nonlinear behavior on an everywhere-defined state space.

Let $n\ge 1$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, let $M:\mathrm{\Omega }\to M_n(\mathbb{R})$ be a $C^2$ field of symmetric positive definite matrices, and let $U:\mathrm{\Omega }\to \mathbb{R}$ be of class $C^2$; let $G$ be the mechanical vector field and $E(q,v)=\frac{1}{2}{v}^{\mathsf{T}}M(q)v+U(q)$ the energy of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart. Let $q_0\in \mathrm{\Omega }$ be a mechanical equilibrium, $\mathrm{\nabla }U(q_0)=0$, and put $K:=D^{2}U(q_0)$ (The Hessian matrix and critical points of a scalar field, The Hessian of a $C^2$ scalar field is symmetric). All chart data, state norms and times are numerical in the fixed reference scales 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' numerical-state convention. Fix also an energy reference $E_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$, so that physical energy is $E_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=E_{\mathrm{r}\mathrm{e}\mathrm{f}}E$ and physical time is $t_{\mathrm{r}\mathrm{e}\mathrm{f}}t$. Then:

1. (Strict minimum.) If $q_0$ is a strict local minimum of $U$, then $(q_0,0)$ is a Lyapunov stable equilibrium of $G$ (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system, 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 (i)).
2. (Linear instability from a negative eigenvalue.) If $K$ has a negative eigenvalue (Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism), then there are $\kappa >0$ and $w\mathrm{\ne }0$ in ${\mathbb{R}}^n$ such that $Y:\mathbb{R}\to {\mathbb{R}}^{2n}$, $Y(t):=e^{\kappa t}(w,\kappa w)$, solves the associated vector equation $Y^{\mathrm{\prime }}=DG(q_0,0)Y$ along the constant solution $z\equiv (q_0,0)$, with $Y(0)=(w,\kappa w)$. This vector equation is the equation for the action on an initial deviation of the normalized matrix equation $A^{\mathrm{\prime }}=DG(q_0,0)A$, $A(0)=I_{2n}$ defined in The variational equation along an ODE solution; in particular $(q_0,0)$ is linearly unstable 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 (iii).
3. (Nonlinear instability from a negative definite Hessian.) If $K$ is negative definite, then every complex eigenvalue of $DG(q_0,0)$ is real and nonzero, at least one is positive, and $(q_0,0)$ is unstable: it is not Lyapunov stable.
4. (No attraction under exact energy conservation.) The energy $E$ is constant along every maximal solution of $G$. If there are states $z=(q,v)$ with $\parallel z-(q_0,0){\parallel }_2$ arbitrarily small and $E(z)\mathrm{\ne }E(q_0,0)$, then $(q_0,0)$ is not asymptotically stable. In particular, if $q_0$ is a strict local minimum of $U$, then $(q_0,0)$ is Lyapunov stable but not asymptotically stable.
5. (Scope.) A zero eigenvalue of $K$ alone gives no stability conclusion; clause 1 still applies to strict minima with degenerate Hessians. Clause 2 gives no nonlinear conclusion when $K$ has both a negative and a positive eigenvalue: the linearisation then also has purely imaginary eigenvalues and is not hyperbolic, so the instability theorem used in clause 3 does not apply. No global stability, no basin of attraction and no statement about the size of any neighbourhood is claimed.

• Why clause 3 needs negative definiteness. The instability theorem of [L2] requires the linearisation to be hyperbolic. Strict concavity alone need not give a negative definite Hessian: $U(q)=-q^4$ is strictly concave but has $U^{\mathrm{\prime }\mathrm{\prime }}(0)=0$. In the rate formulas, ${\lambda }_i$ denotes an eigenvalue of $S=M_0^{-1\mathrm{/}2}K{M}_0^{-1\mathrm{/}2}$ from step 1.2, not of $K$. Negative and positive eigenvalues of $S$ give respectively real rates $\pm \sqrt{-{\lambda }_j}$ and purely imaginary rates $\pm i\sqrt{{\lambda }_k}$; a mixed-sign Hessian therefore leaves centre directions, which the cited hyperbolic theorem does not cover. Clause 2 is the linear statement in that case, and clause 5 records the scope.

• Comparison with the mathematical suppliers. Clause 1 is exactly A strict local minimum of the potential gives a Lyapunov stable equilibrium of the mechanical system, by energy conservation; clause 3 is A hyperbolic equilibrium whose linearisation has an eigenvalue with positive real part is unstable applied after the eigenvalue computations of steps 1.3 and 2.2; clause 4 is the physical reading of Energy is conserved along solutions of the mechanical system with a configuration-dependent kinetic metric and is the reason A strict local minimum of the potential gives a Lyapunov stable equilibrium of the mechanical system, by energy conservation asserts no asymptotic stability.

• Units and coordinates. $[E_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}]=\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework) and the physical rate corresponding to $\kappa$ is $\kappa \mathrm{/}{t}_{\mathrm{r}\mathrm{e}\mathrm{f}}$, of dimension ${\tau }^{-1}$. State norms apply to dimensionless numerical coordinates; fixed positive reference scales preserve the topology and stability predicates. At an equilibrium of a $C^1$ field, a $C^2$ chart transition gives similar derivative matrices, as 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) records.

Let $n\ge 1$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, let $M:\mathrm{\Omega }\to M_n(\mathbb{R})$ be a $C^2$ field of symmetric positive definite matrices, let $U:\mathrm{\Omega }\to \mathbb{R}$ be of class $C^2$, and let $G$ be the mechanical vector field on the state domain $S_0:=\mathrm{\Omega }\times {\mathbb{R}}^n$ (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart). Require the fixed numerical chart convention 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: for declared positive component references $a_j$ and duration reference $t_{\mathrm{r}\mathrm{e}\mathrm{f}}$, physical coordinates and velocities are $Q_j=a_j{q}_j$ and $V_j=(a_j\mathrm{/}{t}_{\mathrm{r}\mathrm{e}\mathrm{f}})v_j$, with physical time $t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=t_{\mathrm{r}\mathrm{e}\mathrm{f}}t$. All data $M,U,G$, balls, vector norms and induced operator norms below are in this dimensionless numerical state chart, so $q^{\mathrm{\prime }}=v$. This field is $C^1$, hence locally Lipschitz; let $\mathrm{\Phi }$ be its local flow as defined in The local flow of an autonomous first-order system, and write $z(t;a)$ for the maximal solution through $a\in S_0$. Fix a state $z_0=(q_0,v_0)\in S_0$, a time $T>0$ and a radius $R>0$ such that $\bar{B}(z_0,R)\subseteq S_0$, put $M_R:={\sup }_{\bar{B}(z_0,R)}\parallel G{\parallel }_2,L_R:={\sup }_{\bar{B}(z_0,R)}\parallel DG\parallel ,$ and let $0<\rho <R-M_{R}T$ (Open ball, closed ball and sphere in a metric space, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case). Then:

1. (Flow derivative and variational equation.) For every $a\in \bar{B}(z_0,\rho )$ the maximal solution $z(\cdot ;a)$ is defined on $[0,T]$ and stays in $B(z_0,R)$; the map $a\mapsto z(t;a)$ is differentiable at every $a\in B(z_0,\rho )$ for $t\in [0,T]$, and for every $u\in {\mathbb{R}}^{2n}$ $D_{a}z(t;a)u=Y_a(t)u,$ where $Y_a:[0,T]\to M_{2n}(\mathbb{R})$ is the unique solution of the variational equation along the reference motion $z(\cdot ;a)$, $Y^{\mathrm{\prime }}(t)=DG(z(t;a))Y(t),Y(0)=I_{2n},$ in the sense of The variational equation along an ODE solution; moreover $\parallel z(t;a)-z(t;b){\parallel }_2\le e^{L_{R}t}\parallel a-b{\parallel }_2$ for all $a,b\in \bar{B}(z_0,\rho )$ and $t\in [0,T]$.
2. (Constant-metric second-order form.) Suppose in addition that $M$ is constant on $\mathrm{\Omega }$, say $M(q)\equiv M$ with $M$ symmetric positive definite. Then, for any $a\in B(z_0,\rho )$, let $q:[0,T]\to \mathrm{\Omega }$ be the position component of its motion $z(t;a)=(q(t),q^{\mathrm{\prime }}(t))$, and write the matrix solution of clause 1 in block form $Y_a(t)=\left(\begin{array}{c}{Y}_{a,q}(t)\\ Y_{a,v}(t)\end{array}\right)$, where each block has $n$ rows and $2n$ columns, and put $\eta (t):=Y_{a,q}(t)$. The variational equation is equivalent, column by column, to the second-order equation $M{\eta }^{\mathrm{\prime }\mathrm{\prime }}(t)=-D^{2}U(q(t))\eta (t)(0\le t\le T),$ with $D^{2}U$ the Hessian of $U$ (The Hessian matrix and critical points of a scalar field).
3. (Conditional finite-time error bound.) Suppose in addition that $z_0$ is an equilibrium, $G(z_0)=0$, and write $A:=DG(z_0)$; suppose there are constants $C\ge 0$, $L\ge 0$ and $r>0$ with $\bar{B}(z_0,r)\subseteq S_0$ and $\parallel G(z)-A(z-z_0){\parallel }_2\le C\parallel z-z_0{\parallel }_2^2,\parallel G(z){\parallel }_2\le L\parallel z-z_0{\parallel }_2(\parallel z-z_0{\parallel }_2\le r)\mathrm{.}$ Let $T>0$ and $\delta :=\parallel a-z_0{\parallel }_2>0$ with $\delta e^{LT}<r$, let $z(\cdot ;a)$ be the model's motion through $a$, and let $y$ solve the linearised equation $y^{\mathrm{\prime }}=Ay$, $y(0)=a-z_0$. Then $\parallel z(t;a)-z_0-y(t){\parallel }_2\le C{\delta }^2{\int }_0^t{e}^{\parallel A\parallel (t-s)}{e}^{2Ls}\mathrm{d}s\le C{\delta }^{2}t{e}^{(\parallel A\parallel +2L)t}(0\le t\le T)\mathrm{.}$ where $\parallel A\parallel$ is the operator norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).
4. (Scope.) The linearised equation approximates the model's motion only for finite times and initial states inside the declared ball with the declared quantitative bounds; the constants $M_R,L_R,C,L,r$ are model-data bounds, not measured quantities. Clause 1 gives no rate of separation beyond the exponential bound, no conclusion when the reference motion leaves $B(z_0,R)$ before time $T$, and no statement about the size of the time interval on which a linearisation is a good approximation; clause 3's hypotheses must be verified for the particular data.