Basic structures of the Shilnikov homoclinic bifurcation scenario

R O Medrano-T, Murilo Da Silva Baptista, I. L. Caldas

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

We find numerically small scale basic structures of homoclinic bifurcation curves in the parameter space of the Chua circuit. The distribution of these basic structures in the parameter space and their geometrical properties constitute a complete homoclinic bifurcation scenario of this system. Furthermore, these structures and the scenario are theoretically demonstrated to be generic to a large class of dynamical systems that presents, as the Chua circuit, Shilnikov homoclinic orbits. We classify the complexity of primary and subsidiary homoclinic orbits by their order given by the number of their returning loops. Our results confirm previous predictions of structures of homoclinic bifurcation curves and extend this study to high order primary orbits. Furthermore, we identify accumulations of bifurcation curves of subsidiary homoclinic orbits into bifurcation curves of both primary and subsidiary orbits. (C) 2005 American Institute of Physics.

Original languageEnglish
Article number033112
Number of pages10
JournalChaos
Volume15
Issue number3
DOIs
Publication statusPublished - Sep 2005

Keywords

  • Volterra-equations
  • strange attractors
  • saddle-focus
  • chaos
  • orbits
  • systems

Cite this

Basic structures of the Shilnikov homoclinic bifurcation scenario. / Medrano-T, R O ; Baptista, Murilo Da Silva; Caldas, I. L.

In: Chaos, Vol. 15, No. 3, 033112, 09.2005.

Research output: Contribution to journalArticle

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AB - We find numerically small scale basic structures of homoclinic bifurcation curves in the parameter space of the Chua circuit. The distribution of these basic structures in the parameter space and their geometrical properties constitute a complete homoclinic bifurcation scenario of this system. Furthermore, these structures and the scenario are theoretically demonstrated to be generic to a large class of dynamical systems that presents, as the Chua circuit, Shilnikov homoclinic orbits. We classify the complexity of primary and subsidiary homoclinic orbits by their order given by the number of their returning loops. Our results confirm previous predictions of structures of homoclinic bifurcation curves and extend this study to high order primary orbits. Furthermore, we identify accumulations of bifurcation curves of subsidiary homoclinic orbits into bifurcation curves of both primary and subsidiary orbits. (C) 2005 American Institute of Physics.

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