attractor
Attractor
2005-05-20 22:45:55
2005-05-20 22:45:55
Definition
attracting set
repelling set
repellor
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Let
$$\dot{x}=f(x)$$
be a system of autonomous ordinary differential equation in $\mathbb{R}^n$ defined by a vector field $f\colon \mathbb{R}^n\to \mathbb{R}^n$.
A set $A$ is said to be an \emph{attracting set}\cite{GH,P} if
\begin{enumerate}
\item $A$ is closed and invariant,
\item there exists an open neighborhood $U$ of $A$ such that all solution
with initial solution in $U$ will eventually enter $A$ ($x(t)\to A$) as $t\to \infty$.
\end{enumerate}
Additionally, if $A$ contains a dense orbit then $A$ is said to be an \emph{attractor}\cite{GH,P}.\\
Conversely, a set $R$ is said to be a \emph{repelling set}\cite{GH} if $R$ satisfy the condition 1. and 2. where $t\to \infty$ is replaced by $t\to -\infty$. Similarly, if $R$ contains a dense orbit then $R$ is said to be a \emph{repellor}\cite{GH}.
\begin{thebibliography}{1}
\bibitem[GH]{GH}
{\scshape Guckenheimer, John \& Holmes, Philip},
\emph{Nonlinear Oscillations, Dynamical Systems,
and Bifurcations of Vector Fields},
Springer, New York, 1983.
\bibitem[P]{P}
{\scshape Perko, Lawrence},
\emph{Differential Equations and Dynamical Systems},
Springer, New York, 2001.
\end{thebibliography}