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Poincaré Map


Consider an n-dimensional deterministic dynamical system

 x^_^.=f^_(x)

and let S be an n-1-dimensional surface of section that is traverse to the flow, i.e., all trajectories starting from S flow through it and are not parallel to it. Then a Poincaré map P is a mapping from S to itself obtained by following trajectories from one intersection of the surface S to the next. Poincaré maps are useful when studying swirling flows near periodic solutions in dynamical systems.


See also

Surface of Section

This entry contributed by Joakim Munkhammar

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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.

Referenced on Wolfram|Alpha

Poincaré Map

Cite this as:

Munkhammar, Joakim. "Poincaré Map." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PoincareMap.html

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