attractor
Let
$$\dot{x}=f(x)$$ 
be a system of autonomous^{} ordinary differential equation^{} in ${\mathbb{R}}^{n}$ defined by a vector field $f:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$. 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 $t\to \mathrm{\infty}$.
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 $t\to \mathrm{\infty}$ is replaced by $t\to \mathrm{\infty}$. 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  20130322 15:17:34 
Last modified on  20130322 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 