Editing Dynamical system (section)

{{Short description|Mathematical model of the time dependence of a point in space}}
{{for|the mechanical concept|Classical dynamics}}
{{Redirect|Dynamical}}
{{More footnotes needed|date=February 2022|partial=y|reason=need at least one per section to tell which reference to look at}}
[[File:Lorenz attractor yb.svg|thumb|right|The [[Lorenz attractor]] arises in the study of the [[Lorenz system|Lorenz oscillator]], a dynamical system.]]

In [[mathematics]], a '''dynamical system''' is a system in which a [[Function (mathematics)|function]] describes the [[time]] dependence of a [[Point (geometry)|point]] in an [[ambient space]], such as in a [[parametric curve]]. Examples include the [[mathematical model]]s that describe the swinging of a clock [[pendulum]], [[fluid dynamics|the flow of water in a pipe]], the [[Brownian motion|random motion of particles in the air]], and [[population dynamics|the number of fish each springtime in a lake]]. The most general definition unifies several concepts in mathematics such as [[ordinary differential equation]]s and [[ergodic theory]] by allowing different choices of the space and how time is measured.{{Citation needed|date=March 2023}} Time can be measured by integers, by [[real number|real]] or [[complex number]]s or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a [[manifold]] or simply a [[Set (mathematics)|set]], without the need of a [[Differentiability|smooth]] space-time structure defined on it.

At any given time, a dynamical system has a [[State (controls)|state]] representing a point in an appropriate [[state space (controls)|state space]]. This state is often given by a [[tuple]] of [[real numbers]] or by a [[vector space|vector]] in a geometrical manifold. The ''evolution rule'' of the dynamical system is a function that describes what future states follow from the current state.  Often the function is [[Deterministic system (mathematics)|deterministic]], that is, for a given time interval only one future state follows from the current state.<ref>{{cite book |last=Strogatz |first=S. H. |year=2001 |title=Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry |publisher=Perseus }}</ref><ref>{{cite book |first1=A. |last1=Katok |first2=B. |last2=Hasselblatt |title=Introduction to the Modern Theory of Dynamical Systems |location=Cambridge |publisher=Cambridge University Press |year=1995 |isbn=978-0-521-34187-5 |url-access=registration |url=https://archive.org/details/introductiontomo0000kato }}</ref>  However, some systems are [[stochastic system|stochastic]], in that random events also affect the evolution of the state variables.

The study of dynamical systems is the focus of ''[[dynamical systems theory]]'', which has applications to a wide variety of fields such as mathematics, physics,<ref>{{cite journal|last1=Melby|first1=P. |display-authors=etal |title=Dynamics of Self-Adjusting Systems With Noise|journal= Chaos: An Interdisciplinary Journal of Nonlinear Science|volume=15 |issue=3 |pages=033902 |date=2005|doi=10.1063/1.1953147|pmid=16252993 |bibcode=2005Chaos..15c3902M}}</ref><ref>{{cite journal|last1=Gintautas|first1=V. |display-authors=etal |title=Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics|journal=J. Stat. Phys. |volume=130|date=2008|issue=3 |page=617 |doi=10.1007/s10955-007-9444-4|arxiv=0705.0311|bibcode=2008JSP...130..617G|s2cid=8677631 }}</ref> [[biology]],<ref>{{cite book |last1=Jackson |first1=T. |last2=Radunskaya |first2=A. |title=Applications of Dynamical Systems in Biology and Medicine |date=2015 |publisher=Springer }}</ref> [[chemistry]], [[engineering]],<ref>{{cite book |first=Erwin |last=Kreyszig |title=Advanced Engineering Mathematics |location=Hoboken |publisher=Wiley |year=2011 |isbn=978-0-470-64613-7 }}</ref> [[economics]],<ref>{{cite book |last=Gandolfo |first=Giancarlo |author-link=Giancarlo Gandolfo |title=Economic Dynamics: Methods and Models |location=Berlin |publisher=Springer |edition=Fourth |year=2009 |orig-year=1971 |isbn=978-3-642-13503-3 }}</ref> [[Cliodynamics|history]], and [[medicine]]. Dynamical systems are a fundamental part of [[chaos theory]], [[logistic map]] dynamics, [[bifurcation theory]], the [[self-assembly]] and [[self-organization]] processes, and the [[edge of chaos]] concept.