Dynamics near a periodically-perturbed robust heteroclinic cycle

Tsung Lung Tsai, Jonathan H.P. Dawes

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Robust heteroclinic cycles (RHCs) arise naturally in collections of symmetric differential equations derived as dynamical models in many fields, including fluid mechanics, game theory and population dynamics. In this paper, we present a careful study of the complicated dynamics generated by small amplitude periodic perturbations of a stable robust heteroclinic cycle (RHC). We give a detailed derivation of the Poincaré map for trajectories near the RHC, asymptotically correct in the limit of small amplitude perturbations. This reduces the nonautonomous system in R3 to a 2D map. We identify three distinct dynamical regimes. The distinctions between these regimes depend subtly on different distinguished limits of the two small parameters in the problem. The first regime corresponds to the RHC being only weakly attracting: here we show that the system is equivalent to a damped nonlinear pendulum with a constant torque. In the second regime the periodically-perturbed RHC is more strongly attracting and the system dynamics corresponds to that of a (non-invertible or invertible) circle map. In the third regime, of yet stronger attraction, the dynamics of the return map is chaotic and no longer reducible to a one-dimensional map. This third regime has been noted previously; our analysis in this paper focusses on providing quantitative results in the first two regimes.

Original languageEnglish
Pages (from-to)14-34
Number of pages21
JournalPhysica D: Nonlinear Phenomena
Volume262
DOIs
Publication statusPublished - 2013 Jan 1

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cycles
game theory
perturbation
fluid mechanics
pendulums
attraction
torque
differential equations
derivation
trajectories

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

Cite this

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Dynamics near a periodically-perturbed robust heteroclinic cycle. / Tsai, Tsung Lung; Dawes, Jonathan H.P.

In: Physica D: Nonlinear Phenomena, Vol. 262, 01.01.2013, p. 14-34.

Research output: Contribution to journalArticle

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