cycle
Cycle4
2005-02-06 20:50:10
2007-01-23 06:58:50
Definition
period
stable cycle
unstable cycle
asymptotically stable cycle
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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 \emph{cycle}\textit{(or \emph{periodic solution})} if it is a closed solution which is not an equilibrium point. The \emph{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 \emph{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 \emph{unstable cycle} if it is not a stable cycle.\\
A cycle, $\Gamma$, is \emph{asymptotically stable cycle} if for all $x\in U$ where $U$ is a neighborhood of $\Gamma$, $\lim_{t\to\infty}d(\phi_t(x),\Gamma)=0$.\cite{PL}\\
\textbf{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 $2\pi$.
\begin{thebibliography}{2}
\bibitem[PL]{PL} Perko, Lawrence: Differential Equations and Dynamical Systems \textit{(Third Edition)}. Springer, New York, 2001.
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