A dynamical system describes how a state evolves with time. A continuous dynamical system is specified by a flow , while a discrete dynamical
system is specified by iterating a map . The state space is
often a manifold or a metric
space . If continuous function , then the evolution
of a variable
(1)
This equation can also be viewed as a difference
equation
(2)
so defining
(3)
gives
(4)
which can be read "as differential
equation
(5)
See also Anosov Diffeomorphism ,
Anosov Flow ,
Axiom
A Diffeomorphism ,
Axiom A Flow ,
Bifurcation
Theory ,
Chaos ,
Entropy
Conjecture ,
Ergodic Theory ,
Geodesic
Flow ,
Mass-Action System Explore this topic in the MathWorld classroom
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References Aoki, N. and Hiraide, K. Topological Theory of Dynamical Systems. Golubitsky,
M. Introduction
to Applied Nonlinear Dynamical Systems and Chaos. Guckenheimer, J. and Holmes, P. Nonlinear
Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed. Jordan, D. W. and Smith, P. Nonlinear
Ordinary Differential Equations: An Introduction to Dynamical Systems, 3rd ed. Lichtenberg, A. and Lieberman,
M. Regular
and Stochastic Motion, 2nd ed. Ott,
E. Chaos
in Dynamical Systems. Rasband,
S. N. Chaotic
Dynamics of Nonlinear Systems. Strogatz,
S. H. Nonlinear
Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering. Tabor, M. Chaos
and Integrability in Nonlinear Dynamics: An Introduction. Referenced on Wolfram|Alpha Dynamical System
Cite this as:
Weisstein, Eric W. "Dynamical System."
From MathWorld https://mathworld.wolfram.com/DynamicalSystem.html
Subject classifications
is any continuous function, then the evolution
of a variable 
can be given by the formula
changes by 1 unit, 
changes by 
." This is the discrete analog of the differential
equation
