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 is any continuous function, then the evolution of a variable can be given by the formula

This equation can also be viewed as a difference equation

so defining

gives

which can be read "as changes by 1 unit, changes by ." This is the discrete analog of the differential equation

where is the derivative of with respect to continuous time .

A continuous example is the system defined by Chua's circuit equations, which can exhibit a double-scroll attractor with two spiral-shaped lobes.

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. 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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Dynamical System

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Weisstein, Eric W. "Dynamical System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DynamicalSystem.html

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