The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets

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.