Dynamical systems theory

Dynamical systems theory is an area of <a href="/facts/Mathematics/pxTouaz4">mathematics</a> used to describe the behavior of <a href="/facts/Complex_systems/eGG3B1nn">complex</a> <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a>, usually by employing <a href="/facts/Differential_equations/9MGbE3Ca">differential equations</a> or <a href="/facts/Difference_equations/JolXC71M">difference equations</a>. When differential equations are employed, the theory is called <a href="/facts/Continuous_time/RR8wpS9e">continuous dynamical systems</a>. From a physical point of view, continuous dynamical systems is a generalization of <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a>, a generalization where the <a href="/facts/Equations_of_motion/25msr1vE">equations of motion</a> are postulated directly and are not constrained to be <a href="/facts/Euler%E2%80%93Lagrange_equation/Ymq3r3Dh">Euler–Lagrange equations</a> of a <a href="/facts/Principle_of_least_action/QaBTpoKz">least action principle</a>. When difference equations are employed, the theory is called <a href="/facts/Discrete_time/RR8wpS9e">discrete dynamical systems</a>. 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 <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a>, one gets <a href="/facts/Dynamic_equations_on_time_scales/9T4xVS0t">dynamic equations on time scales</a>. Some situations may also be modeled by mixed operators, such as <a href="/facts/Differential-difference_equations/fM5wzXKr">differential-difference equations</a>.
This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the <a href="/facts/Equations_of_motion/25msr1vE">equations of motion</a> of systems that are often primarily <a href="/facts/Mechanics/A3yHLQA5">mechanical</a> or otherwise physical in nature, such as <a href="/facts/Planetary_orbit/nD2TPySG">planetary orbits</a> and the behaviour of <a href="/facts/Electronic_circuit/IRfQ6Cc2">electronic circuits</a>, as well as systems that arise in <a href="/facts/Biology/1M1ee3pB">biology</a>, <a href="/facts/Economics/HYctA2Dn">economics</a>, and elsewhere. Much of modern research is focused on the study of <a href="/facts/Chaotic_system/LTC17vsF">chaotic systems</a> and bizarre systems.
This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.

Dynamical systems theory open-in-new

Dynamical systems theory