# 2.8] What is an attractor? (nonlinear science)

## Description

This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by
others.

# 2.8] What is an attractor? (nonlinear science)

Informally an attractor is simply a state into which a system settles (thus

dissipation is needed). Thus in the long term, a dissipative dynamical system

may settle into an attractor.

Interestingly enough, there is still some controversy in the mathematics

community as to an appropriate definition of this term. Most people adopt the

definition

Attractor: A set in the phase space that has a neighborhood in which every

point stays nearby and approaches the attractor as time goes to infinity.

Thus imagine a ball rolling inside of a bowl. If we start the ball at a point

in the bowl with a velocity too small to reach the edge of the bowl, then

eventually the ball will settle down to the bottom of the bowl with zero

velocity: thus this equilibrium point is an attractor. The neighborhood of

points that eventually approach the attractor is the basin of attraction for

the attractor. In our example the basin is the set of all configurations

corresponding to the ball in the bowl, and for each such point all small

enough velocities (it is a set in the four dimensional phase space [2.4]).

Attractors can be simple, as the previous example. Another example of an

attractor is a limit cycle, which is a periodic orbit that is attracting

(limit cycles can also be repelling). More surprisingly, attractors can be

chaotic (see [2.9]) and/or strange (see [2.12]).

The boundary of a basin of attraction is often a very interesting object

since it distinguishes between different types of motion. Typically a basin

boundary is a saddle orbit, or such an orbit and its stable manifold. A crisis

is the change in an attractor when its basin boundary is destroyed.

An alternative definition of attractor is sometimes used because there

are systems that have sets that attract most, but not all, initial conditions

in their neighborhood (such phenomena is sometimes called riddling of the

basin). Thus, Milnor defines an attractor as a set for which a positive

measure (probability, if you like) of initial conditions in a neighborhood are

asymptotic to the set.

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