# cycle

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 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

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

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

 $\displaystyle x(t)$ $\displaystyle=$ $\displaystyle\cos t$ $\displaystyle y(t)$ $\displaystyle=$ $\displaystyle\sin t$

which has a period of $2\pi$.

## References

 Title cycle Canonical name Cycle12 Date of creation 2013-03-22 15:00:51 Last modified on 2013-03-22 15:00:51 Owner Daume (40) Last modified by Daume (40) Numerical id 6 Author Daume (40) Entry type Definition Classification msc 34A12 Classification msc 34C07 Synonym periodic solution Synonym stable periodic solution Synonym unstable periodic solution Synonym asymptotically stable periodic solution Defines period Defines stable cycle Defines unstable cycle Defines asymptotically stable cycle