Symbolic computation of secondary bifurcations in a parametrically excited simple pendulum

Eric A. Butcher, S. C. Sinha

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

A symbolic computational technique is used to study the secondary bifurcations of a parametrically excited simple pendulum as an explicit function of the periodic parameter. This is made possible by the recent development of an algorithm which approximates the fundamental solution matrix of linear time-periodic systems in terms of system parameters in symbolic form. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet transition matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation boundaries in the parameter space. Since this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. By repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution, it is shown how the bifurcation locations as well as the new linearized equations may be obtained in closed form as a function of the periodic parameter. Bifurcation diagrams are constructed and the results are compared with those obtained elsewhere using the point mapping method.

Original languageEnglish (US)
Pages (from-to)627-637
Number of pages11
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume8
Issue number3
DOIs
StatePublished - Mar 1998

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics

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