Poincaré Map

Consider an -dimensional deterministic dynamical system

and let  be an -dimensional surface of section that is traverse to the flow, i.e., all trajectories starting from  flow through it and are not parallel to it. Then a Poincaré map  is a mapping from  to itself obtained by following trajectories from one intersection of the surface  to the next. Poincaré maps are useful when studying swirling flows near periodic solutions in dynamical systems.

REFERENCES:

Devaney, R. L. An Introduction to Chaotic Dynamical Systems, 2nd ed. Westview Press, 2003.

Gleick, J. Chaos: Making a New Science. New York: Penguin Books, p. 142, 1988.

Rasband, S. N. "The Poincaré Map." §5.3 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 92-95, 1990.

Strogatz, S. H. Nonlinear Dynamics and Chaos. Perseus Books, 1994.