Dynamical system

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a dynamical system is a system in which a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> describes the <a href="/facts/Time/3ShdpqLx">time</a> dependence of a <a href="/facts/Point_(geometry)/6YPAvX2a">point</a> in an <a href="/facts/Ambient_space/umgCRklf">ambient space</a>, such as in a <a href="/facts/Parametric_curve/5Uvrlgqf">parametric curve</a>. Examples include the <a href="/facts/Mathematical_model/CGhyuxgz">mathematical models</a> that describe the swinging of a clock <a href="/facts/Pendulum/D2hQXwlI">pendulum</a>, <a href="/facts/Fluid_dynamics/y7uzSCfg">the flow of water in a pipe</a>, the <a href="/facts/Brownian_motion/Th0g0wkr">random motion of particles in the air</a>, and <a href="/facts/Population_dynamics/WXbtbncd">the number of fish each springtime in a lake</a>. The most general definition unifies several concepts in mathematics such as <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a> and <a href="/facts/Ergodic_theory/rkqCbIy0">ergodic theory</a> by allowing different choices of the space and how time is measured. Time can be measured by integers, by <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a <a href="/facts/Manifold/rXPERJtu">manifold</a> or simply a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>, without the need of a <a href="/facts/Differentiability/jQqTgLk1">smooth</a> space-time structure defined on it.
At any given time, a dynamical system has a <a href="/facts/State_(controls)/Jf4EanCP">state</a> representing a point in an appropriate <a href="/facts/State_space_(controls)/Jf4EanCP">state space</a>. This state is often given by a <a href="/facts/Tuple/BI1HIxL8">tuple</a> of <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a> or by a <a href="/facts/Vector_space/M1pxMLx2">vector</a> 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 <a href="/facts/Deterministic_system_(mathematics)/oT1NvyHD">deterministic</a>, that is, for a given time interval only one future state follows from the current state. However, some systems are <a href="/facts/Stochastic_system/Ng1n1dnB">stochastic</a>, in that random events also affect the evolution of the state variables.
The study of dynamical systems is the focus of <a href="/facts/Dynamical_systems_theory/LM6aDhV0">dynamical systems theory</a>, which has applications to a wide variety of fields such as mathematics, physics, <a href="/facts/Biology/1M1ee3pB">biology</a>, <a href="/facts/Chemistry/SrNqkGVm">chemistry</a>, <a href="/facts/Engineering/RFVg8H3I">engineering</a>, <a href="/facts/Economics/HYctA2Dn">economics</a>, <a href="/facts/Cliodynamics/rx3NQxFv">history</a>, and <a href="/facts/Medicine/2fnp90Je">medicine</a>. Dynamical systems are a fundamental part of <a href="/facts/Chaos_theory/LTC17vsF">chaos theory</a>, <a href="/facts/Logistic_map/wskv7mRQ">logistic map</a> dynamics, <a href="/facts/Bifurcation_theory/sLsN54Rn">bifurcation theory</a>, the <a href="/facts/Self-assembly/HgFyhl6q">self-assembly</a> and <a href="/facts/Self-organization/NSGBKxUa">self-organization</a> processes, and the <a href="/facts/Edge_of_chaos/mDvKcs7z">edge of chaos</a> concept.

Dynamical system open-in-new

Dynamical system