Editing Dynamical systems theory (section)

{{Short description|Area of mathematics used to describe the behavior of complex dynamical systems}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}
'''Dynamical systems theory''' is an area of [[mathematics]] used to describe the behavior of [[complex systems|complex]] [[dynamical system]]s, usually by employing [[differential equations]] by nature of the [[ergodic theory|ergodicity]] of dynamic systems. When differential equations are employed, the theory is called [[continuous time|''continuous dynamical systems'']]. From a physical point of view, continuous dynamical systems is a generalization of [[classical mechanics]], a generalization where the [[equations of motion]] are postulated directly and are not constrained to be [[Euler–Lagrange equation]]s of a [[Principle of least action|least action principle]]. When difference equations are employed, the theory is called [[discrete time|''discrete dynamical systems'']]. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a [[Cantor set]], one gets [[dynamic equations on time scales]]. Some situations may also be modeled by mixed operators, such as [[differential-difference equations]].

This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the [[equations of motion]] of systems that are often primarily [[mechanics|mechanical]] or otherwise physical in nature, such as [[planetary orbit]]s and the behaviour of [[electronic circuit]]s, as well as systems that arise in [[biology]], [[economics]], and elsewhere. Much of modern research is focused on the study of [[chaotic system]]s and bizarre systems.

This field of study is also called just ''dynamical systems'', ''mathematical dynamical systems theory'' or the ''mathematical theory of dynamical systems''.
[[Image:Lorenz attractor yb.svg|thumb|240px|right|A chaotic solution of the [[Lorenz system]], which is an example of a [[non-linear]] dynamical system. Studying the Lorenz system helped give rise to [[chaos theory]].]]