attractor

Let

$$\dot{x}=f(x)$$

be a system of autonomous ordinary differential equation in ${ℝ}^n$ defined by a vector field $f:{ℝ}^n\to {ℝ}^n$.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 solutionwith 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].