Editing Dynamical systems theory (section)

== Related fields ==

=== Arithmetic dynamics ===
:[[Arithmetic dynamics]] is a field that emerged in the 1990s that amalgamates two areas of mathematics, [[dynamical systems]] and [[number theory]]. Classically, discrete dynamics refers to the study of the [[Iterated function|iteration]] of self-maps of the [[complex plane]] or [[real line]]. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, {{math|<var>p</var>}}-adic, and/or algebraic points under repeated application of a [[polynomial]] or [[rational function]].

=== Chaos theory ===
:[[Chaos theory]] describes the behavior of certain [[dynamical system (definition)|dynamical system]]s – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the [[butterfly effect]]). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears [[randomness|random]]. This happens even though these systems are [[deterministic system (philosophy)|deterministic]], meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply ''chaos''.

=== Complex systems ===
:[[Complex systems]] is a scientific field that studies the common properties of [[system]]s considered [[Complexity|complex]] in [[nature]], [[society]], and [[science]]. It is also called ''complex systems theory'', ''complexity science'', ''study of complex systems'' and/or ''sciences of complexity''. The key problems of such systems are  difficulties with their formal [[Scientific modelling|modeling]] and [[simulation]]. From such perspective, in different research contexts complex systems are defined on the base of their different attributes.

:The study of complex systems is bringing new vitality to many areas of science where a more typical [[reductionist]] strategy has fallen short. ''Complex systems'' is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including [[neurosciences]], [[social sciences]], [[meteorology]], [[chemistry]], [[physics]], [[computer science]], [[psychology]], [[artificial life]], [[evolutionary computation]], [[economics]], earthquake prediction, [[molecular biology]] and inquiries into the nature of living [[cell (biology)|cell]]s themselves.

=== Control theory ===
:[[Control theory]] is an interdisciplinary branch of [[engineering]] and [[mathematics]], in part it deals with influencing the behavior of [[dynamical system]]s.

=== Ergodic theory ===
:[[Ergodic theory]] is a branch of [[mathematics]] that studies [[dynamical system]]s with an [[invariant measure]] and related problems. Its initial development was motivated by problems of [[statistical physics]].

=== Functional analysis ===
:[[Functional analysis]] is the branch of [[mathematics]], and specifically of [[mathematical analysis|analysis]], concerned with the study of [[vector space]]s and [[operator (mathematics)|operator]]s acting upon them. It has its historical roots in the study of [[functional space]]s, in particular transformations of [[function (mathematics)|functions]], such as the [[Fourier transform]], as well as in the study of [[differential equations|differential]] and [[integral equations]]. This usage of the word ''[[functional (mathematics)|functional]]'' goes back to the [[calculus of variations]], implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist [[Vito Volterra]] and its founding is largely attributed to mathematician [[Stefan Banach]].

=== Graph dynamical systems ===
:The concept of [[graph dynamical system]]s (GDS) can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of graph dynamical systems is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.

=== Projected dynamical systems ===
:[[Projected dynamical systems]] is a [[mathematics|mathematical]] theory investigating the behaviour of [[dynamical system]]s where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of [[Optimization (mathematics)|optimization]] and [[Equilibrium point|equilibrium]] problems and the dynamical world of [[ordinary differential equations]]. A projected dynamical system is given by the [[flow (mathematics)|flow]] to the projected differential equation.

=== Symbolic dynamics ===
:[[Symbolic dynamics]] is the practice of modelling a topological or smooth [[dynamical system]] by a discrete space consisting of infinite [[sequence]]s of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the [[shift operator]].

=== System dynamics ===
:[[System dynamics]] is an approach to understanding the behaviour of systems over time.  It deals with internal feedback loops and time delays that affect the behaviour and state of the entire system.<ref name="sysdyn">[http://sysdyn.clexchange.org MIT System Dynamics in Education Project (SDEP)<!-- Bot generated title -->] {{webarchive|url=https://web.archive.org/web/20080509163801/http://sysdyn.clexchange.org/ |date=2008-05-09 }}</ref>  What makes using system dynamics different from other approaches to studying systems is the language used to describe [[feedback]] loops with [[Stock and flow|stocks and flows]].  These elements help describe how even seemingly simple systems display baffling [[nonlinearity]].

=== Topological dynamics ===
:[[Topological dynamics]] is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of [[general topology]].