Hyperbolic equilibria, hyperbolic fixed points, periodic orbits, limit cycles and homoclinic trajectories

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and let $F:U\to {\mathbb{R}}^n$ be a map; a solution means a solution of $z^{\mathrm{\prime }}(t)=F(z(t))$ as in Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system.

For a real linear map $A:{\mathbb{R}}^n\to {\mathbb{R}}^n$, define its complex-linear extension $A_{\mathbb{C}}:{\mathbb{C}}^n\to {\mathbb{C}}^n$ by $A_{\mathbb{C}}(x+iy):=Ax+iAy$ for $x,y\in {\mathbb{R}}^n$. In both spectral definitions below, an eigenvalue means an eigenvalue of $A_{\mathbb{C}}$ over $\mathbb{C}$ (Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism with field $F=\mathbb{C}$); $\mathrm{Re}\lambda$ and $\mathrm{\mid }\lambda \mathrm{\mid }$ are the real part and modulus of Real and imaginary parts, complex conjugation, and modulus.

Hyperbolic equilibrium. Let $z_{\ast }\in U$ be an equilibrium of $F$ and suppose $F$ is totally differentiable at $z_{\ast }$ (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder) with total derivative $DF(z_{\ast })$. Then $z_{\ast }$ is hyperbolic when no eigenvalue of the real matrix of $DF(z_{\ast })$ has real part $0$ (Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism).

Hyperbolic fixed point of a map. Let $V\subseteq {\mathbb{R}}^n$ be open, let $\phi :V\to V$ be a map and let $p\in V$ satisfy $\phi (p)=p$. Suppose $\phi$ is totally differentiable at $p$. Then $p$ is a hyperbolic fixed point of $\phi$ when no eigenvalue of $D\phi (p)$ has modulus $1$.

Periodic solutions and periodic orbits. A solution $z:\mathbb{R}\to U$ is $T$-periodic for $T>0$ when $z(t+T)=z(t)$ for every $t\in \mathbb{R}$, and periodic when it is $T$-periodic for some $T>0$. It is nonconstant when $z(t_1)\mathrm{\ne }z(t_2)$ for some $t_1,t_2$. The image of a periodic solution is $\{z(t):t\in \mathbb{R}\}$; a subset of $U$ is a periodic orbit when it is the image of a nonconstant periodic solution. A number $T>0$ is a least period of $z$ when $T$ is a period of $z$ and no $T^{\mathrm{\prime }}\in (0,T)$ is a period of $z$.

Limit cycle. A periodic orbit $S\subseteq U$ is a limit cycle when it is isolated: there is an open set $W\subseteq U$ with $S\subseteq W$ such that the image of every nonconstant periodic solution lying in $W$ equals $S$; equivalently, no other periodic orbit of the system is contained in $W$. Stability or attraction of $S$ is a separate property and is not part of the definition.

Homoclinic and heteroclinic trajectories. A solution $z$ defined on all of $\mathbb{R}$ is homoclinic to the equilibrium $z_{\ast }$ when $z(t)\to z_{\ast }$ as $t\to +\mathrm{\infty }$ and as $t\to -\mathrm{\infty }$ and $z$ is not the constant solution $z\equiv z_{\ast }$. It is heteroclinic from $z_{\ast }$ to $z_{\ast \ast }$ when $z_{\ast }\mathrm{\ne }{z}_{\ast \ast }$ are equilibria and $z(t)\to z_{\ast \ast }$ as $t\to +\mathrm{\infty }$ and $z(t)\to z_{\ast }$ as $t\to -\mathrm{\infty }$.

• Hyperbolicity is a spectral condition on the derivative at a single point. It does not by itself assert Lyapunov stability, instability, the existence of local stable or unstable manifolds, or any statement about the nonlinear system; each of those is a theorem about the system under stated hypotheses, not a definitional consequence.

• The qualifier "isolated" in the definition of a limit cycle is taken with respect to the periodic orbits of the same system on an open neighbourhood of the cycle's image. A given periodic orbit may be isolated in this sense while being orbitally unstable; conversely an orbitally stable periodic orbit in a system with a continuum of periodic orbits filling an annulus is not a limit cycle.

• Stability and attraction of an invariant set are defined in Invariant sets, orbital stability of an invariant set, and linear stability of an equilibrium; for a periodic orbit they are separate from the isolation condition used here, and neither is implied by it.

• For an equilibrium $z_{\ast }$, adopt the all-solutions convention when uniqueness is not assumed. For $a\in U$, let $\mathcal{M}(a)$ be the set of maximal solutions through $z(0)=a$, with maximality as in Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system. Define $W^{+}(z_{\ast })$ to consist of those $a$ for which $\mathcal{M}(a)\mathrm{\ne }\mathrm{\varnothing }$ and every $z:J\to U$ in $\mathcal{M}(a)$ has $[0,\mathrm{\infty })\subseteq J$ and $z(t)\to z_{\ast }$ as $t\to +\mathrm{\infty }$. Define $W^{-}(z_{\ast })$ by the same nonemptiness requirement, $(-\mathrm{\infty },0]\subseteq J$ for every such solution, and convergence as $t\to -\mathrm{\infty }$. With a unique maximal solution these reduce to the usual stable and unstable sets; without uniqueness they explicitly require convergence of all trajectories and differ from an existential convention. Calling either set a differentiable manifold requires a stable-manifold theorem with its own hypotheses; no such conclusion is drawn here.