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Succinct representation of the Poincare map for periodically driven differential equations

Deane, Jonathan H. B. and Jefferies, David J. (1998) Succinct representation of the Poincare map for periodically driven differential equations Complexity International, 6.


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A differential equation, periodically driven with period T, defines the time evolution of the solution, a state vector x(t). The Poincaré, or time one, map is a function that relates to x(t + T)to x(t). For most second and higher order nonlinear differential equations, the Poincaré map is not available in a closed form; it can generally only be inferred from numerical calculations.

In this paper, we derive an iterative representation of the Poincaré map for Duffing's equation . Our objectives are (a) to represent the mapping in as succinct a form as possible (compact enough to be published in this paper) and (b) to demonstrate that this map representation adequately reproduces the behaviour of Duffing's equation, for instance bifurcation diagrams, co-existing attractors and Poincaré sections. We succeed in these objectives, and our representation increases computation speed by a factor of 45 over traditional numerical calculations.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Deane, Jonathan H.
Jefferies, David J.
Date : 15 October 1998
Additional Information : Published in Complexity International, Vol. 6. Published by Monash University. This material is the work of the author/s listed here. It may be freely copied and distributed provided that the names of the authors and of the journal remain attached.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:40
Last Modified : 16 Jan 2019 16:21

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