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

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.