# 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. 1.

$A$ is closed and invariant,

2. 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

Title attractor Attractor 2013-03-22 15:17:34 2013-03-22 15:17:34 Daume (40) Daume (40) 4 Daume (40) Definition msc 34C99 attracting set repelling set repellor