Editing Attractor (section)

== Motivation of attractors ==
A [[dynamical system]] is generally described by one or more [[differential equations|differential]] or [[difference equations]]. The equations of a given dynamical system specify its behavior over any given short period of time. To determine the system's behavior for a longer period, it is often necessary to [[Integral|integrate]] the equations, either through analytical means or through [[iteration]], often with the aid of computers.

Dynamical systems in the physical world tend to arise from [[dissipative system]]s: if it were not for some driving force, the motion would cease.  (Dissipation may come from [[friction|internal friction]], [[thermodynamics|thermodynamic losses]], or loss of material, among many causes.) The dissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior. The subset of the [[phase space]] of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.

Invariant sets and [[limit set]]s are similar to the attractor concept. An ''invariant set'' is a set that evolves to itself under the dynamics.<ref>{{cite book|author1=Carvalho, A.|author2=Langa, J.A.|author3=Robinson, J.| year=2012|title=Attractors for infinite-dimensional non-autonomous dynamical systems|volume=182|publisher=Springer|page=109}}</ref> Attractors may contain invariant sets. A ''limit set'' is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.

For example, the [[damping ratio|damped]] [[pendulum]] has two invariant points: the point {{math|''x''<sub>0</sub>}} of minimum height and the point {{math|''x''<sub>1</sub>}} of maximum height. The point {{math|x<sub>0</sub>}} is also a limit set, as trajectories converge to it; the point {{math|''x''<sub>1</sub>}} is not a limit set.  Because of the dissipation due to air resistance, the point {{math|x<sub>0</sub>}} is also an attractor.  If there was no dissipation, {{math|''x''<sub>0</sub>}} would not be an attractor. Aristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.

Some attractors are known to be chaotic (see [[#Strange attractor|strange attractor]]), in which case the evolution of any two distinct points of the attractor result in exponentially [[chaos theory|diverging trajectories]], which complicates prediction when even the smallest noise is present in the system.<ref>{{cite book|author1=Kantz, H.|author2=Schreiber, T.|year=2004|title=Nonlinear time series analysis|publisher=Cambridge university press}}</ref>