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# 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*(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

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

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

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

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

## Mathematics Subject Classification

34A12*no label found*34C07

*no label found*

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