Let $$\dot{x}=f(x)$$ be an autonomous ordinary differential equation defined by the vector field $f\colon V \to V$ 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 $\phi_t(x)$ be the flow defined by the above ODE and $d$ be the metric of $V$ then:
A cycle, $\Gamma$ , is a stable cycle if for all $\epsilon>0$ there exists a neighborhood $U$ of $\Gamma$ such that for all $x\in U$ , $d(\phi_t(x),\Gamma)< \epsilon$ .
A cycle, $\Gamma$ , is unstable cycle if it is not a stable cycle.
A cycle, $\Gamma$ , is asymptotically stable cycle if for all $x\in U$ where $U$ is a neighborhood of $\Gamma$ , $\lim_{t\to\infty}d(\phi_t(x),\Gamma)=0$ .[PL]

example:
Let \begin{eqnarray*} \dot{x} & = & -y\\ \dot{y} & = & x \end{eqnarray*}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 \begin{eqnarray*} x(t) & = & \cos t\\ y(t) & = & \sin t \end{eqnarray*}which has a period of $2\pi$ .

References

PL

Perko, Lawrence: Differential Equations and Dynamical Systems (Third Edition). Springer, New York, 2001.