Dynamical systems theory

Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set which is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set then you get dynamic equations on time scales. Some situations may also be modelled by mixed operators such as differential-difference equations.

This theory deals with the long-term qualitative behavior of dynamical systems, and the studies of the solutions to the equations of motion of systems that are primarily mechanical in nature; although this includes both planetary orbits as well as the behaviour of electronic circuits and the solutions to partial differential equations that arise in biology. Much of modern research is focused on the study of chaotic systems.

This field of study is also called just Dynamical systems, Systems theory or longer as Mathematical Dynamical Systems Theory and the Mathematical theory of dynamical systems.

The Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to Chaos theory.

Contents

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• 1 Overview
• 2 History
• 3 Concepts
  • 3.1 Dynamical systems
  • 3.2 Dynamicism
  • 3.3 Nonlinear system
• 4 Related fields
  • 4.1 Arithmetic dynamics
  • 4.2 Chaos theory
  • 4.3 Complex systems
  • 4.4 Control theory
  • 4.5 Ergodic theory
  • 4.6 Functional analysis
  • 4.7 Graph dynamical systems
  • 4.8 Projected dynamical systems
  • 4.9 Symbolic dynamics
  • 4.10 System dynamics
  • 4.11 Topological dynamics
• 5 Applications
  • 5.1 In biomechanics
  • 5.2 In cognitive science
• 6 See also
• 7 Notes
• 8 Further reading
• 9 External links