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Dynamical System

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

x_(n+1)=f(x_n).
(1)

This equation can also be viewed as a difference equation

x_(n+1)-x_n=f(x_n)-x_n,
(2)

so defining

g(x)=f(x)-x
(3)

gives

x_(n+1)-x_n=g(x_n)*1,
(4)

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

x^'(n)=g(x(n)).
(5)

See also

Anosov Diffeomorphism, Anosov Flow, Axiom A Diffeomorphism, Axiom A Flow, Bifurcation Theory, Chaos, Ergodic Theory, Geodesic Flow, Symbolic Dynamics 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

Cite this as:

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

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