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  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)

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.