Succinct representation of the Poincare map for periodically driven differential equations

Abstract

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.

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