Local periodic-orbit, Poincare-return, and Floquet data

For a real $C^1$ vector field $\dot{x}=f(x)$ on an open subset of ${\mathbb{R}}^d$, a nonconstant solution $x_{\ast }(t)$ with least positive period $T$ has periodic orbit $\mathrm{\Gamma }=\{x_{\ast }(t):0\le t<T\}$. Its variational fundamental matrix $Y(t)$ solves

$\dot{Y}(t)=Df(x_{\ast }(t))Y(t),Y(0)=I\mathrm{.}$

The matrix $Y(T)$ is the monodromy matrix; its eigenvalues, counted with algebraic multiplicity, are the Floquet multipliers. The multiplier $1$ in the direction ${\dot{x}}_{\ast }(0)$ is called the phase multiplier.

A local Poincare section at $x_{\ast }(0)$ is a smooth codimension-one surface through that point whose tangent space does not contain $f(x_{\ast }(0))$. The local return map, on a sufficiently small section neighborhood where the next return is defined, sends a point to its next intersection after one turn around $\mathrm{\Gamma }$. Its derivative at $x_{\ast }(0)$ is the Poincare derivative.

The orbit is orbitally asymptotically stable if it is Lyapunov stable as a set and the distance from every sufficiently nearby trajectory to $\mathrm{\Gamma }$ tends to zero as time tends to $+\mathrm{\infty }$. It is orbitally unstable if it is not Lyapunov stable as a set.