attractor

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 attracting set[GH, P] if

1.

$A$ is closed and invariant,
2.

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 $t\to\infty$ .

Additionally, if $A$ contains a dense orbit then $A$ is said to be an attractor[GH, P].
Conversely, a set $R$ is said to be a repelling set[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 repellor[GH].

References

GH Guckenheimer, John & Holmes, Philip, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
P Perko, Lawrence, Differential Equations and Dynamical Systems, Springer, New York, 2001.

Title	attractor
Canonical name	Attractor
Date of creation	2013-03-22 15:17:34
Last modified on	2013-03-22 15:17:34
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	4
Author	Daume (40)
Entry type	Definition
Classification	msc 34C99
Defines	attracting set
Defines	repelling set
Defines	repellor