cycle

Let

$$\dot{x}=f(x)$$

be an autonomous ordinary differential equation defined by the vector field $f:V\to V$ then $x(t)\in V$ a solution of the system is a cycle(or periodic solution) if it is a closed solution which is not an equilibrium point. The period of a cycle is the smallest positive $T$ such that $x(t)=x(t+T)$.
Let ${\varphi }_t(x)$ be the flow defined by the above ODE and $d$ be the metric of $V$ then:
A cycle, $\mathrm{\Gamma }$, is a stable cycle if for all $ϵ>0$ there exists a neighborhood $U$ of $\mathrm{\Gamma }$ such that for all $x\in U$, .
A cycle, $\mathrm{\Gamma }$, is unstable cycle if it is not a stable cycle.
A cycle, $\mathrm{\Gamma }$, is asymptotically stable cycle if for all $x\in U$ where $U$ is a neighborhood of $\mathrm{\Gamma }$, ${lim}_{t\to \mathrm{\infty }}d({\varphi }_t(x),\mathrm{\Gamma })=0$.[PL]

example:
Let

$\dot{x}$$=$$-y$

$\dot{y}$$=$$x$

then the above autonomous ordinary differential equations with initial value condition $(x(0),y(0))=(1,0)$ has a solution which is a stable cycle. Namely the solution defined by

$x(t)$$=$$\cos t$

$y(t)$$=$$\sin t$

which has a period of $2\pi$.