Succinct representation of the Poincare ( ¶ ) map for periodically driven differential equations

J.H.B. Deane and D.J. Jefferies
Department of Electronic and Electrical Engineering,
University of Surrey,
Guildford GU2 5XH,
United Kingdom
e-mail: ees1jd/ees1dj@ee.surrey.ac.uk

Abstract:

A differential equation, periodically driven with period T, defines the time evolution of the solution, a state vector . The ¶, or time one, map is a function that relates to . For most second and higher order nonlinear differential equations, the ¶ 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 ¶ 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 and ¶ sections. We succeed in these objectives, and our representation increases computation speed by a factor of 45 over traditional numerical calculations.

• Introduction
• Derivation of the representation
  • Analytical continuation
  • The recursion formula
• Results
  • Basin of attraction
  • Comparison of ¶ sections
  • Comparison of bifurcation diagrams
• Discussion and conclusions
• Appendix
• References
• About this document ...