4.1 Periodic Orbits

Periodic Orbit

Given the dynamic system $\dot{x} = F (x)$ , the flow $ϕ^{t} (x_{0})$ is a periodic orbit with minimal period $T$ , if:

\begin{aligned} ϕ^{T} (x_{0}) & = x_{0} \\ ϕ^{T} (x_{0}) & \neq ϕ (x_{0}) for any 0 < t < T \end{aligned}

Limit Cycle

Is an isolated periodic orbit

Poincaré-Bendixon

Dynamic system $\dot{x} = F (x)$ in $R^{2}$ .
If $A$ is compact and is positively invariant for the system. Then for any point $x_{0} \in A$ , Either:

$ω (x)$ consists of a single fixed point
$ω (x)$ consists of a single periodic orbit
$ω (x)$ consists of several (but finite) fixed points connected by homoclinic/heteroclinic orbits.

Finding Periodic Orbit

show that $A$ is closed
show that $A$ is bounded
show that $A$ is positively invariant
show that there exists no fixed point in $A$