Dynamical System -- from Wolfram MathWorld
- ️Weisstein, Eric W.
A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers
on another object (usually a manifold). When the reals
are acting, the system is called a continuous dynamical system, and when the integers
are acting, the system is called a discrete dynamical system. If is any continuous function,
then the evolution of a variable
can be given by the formula
(1) |
This equation can also be viewed as a difference equation
(2) |
so defining
(3) |
gives
(4) |
which can be read "as changes by 1 unit,
changes by
." This is the discrete analog of the differential
equation
(5) |
See also
Anosov Diffeomorphism, Anosov Flow, Axiom A Diffeomorphism, Axiom A Flow, Bifurcation Theory, Chaos, Ergodic Theory, Geodesic Flow Explore this topic in the MathWorld classroom
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References
Aoki, N. and Hiraide, K. Topological Theory of Dynamical Systems. Amsterdam, Netherlands: North-Holland, 1994.Golubitsky, M. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, 1997.Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed. New York: Springer-Verlag, 1997.Jordan, D. W. and Smith, P. Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, 3rd ed. Oxford, England: Oxford University Press, 1999.Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion, 2nd ed. New York: Springer-Verlag, 1994.Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993.Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, 1990.Strogatz, S. H. Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering. Reading, MA: Addison-Wesley, 1994.Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.
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Cite this as:
Weisstein, Eric W. "Dynamical System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DynamicalSystem.html