Homoclinic orbits in a piecewise system and their relation with invariant sets

R O Medrano-T, M S Baptista, I L Caldas

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20 Citations (Scopus)


Basic phenomena in chaos can be associated with homoclinic and heteroclinic orbits. In this paper, we present a general numerical method to demonstrate the existence of these orbits in piecewise-linear systems. We also show that the tangency of the stable and unstable manifolds, at the onset of the chaotic double-scroll attractor, changes the basin boundaries of two alpha-limit sets. These changes are evidence of homoclinicity in the dynamical system. These basins give complete information about the stable manifolds around the fixed points. We show that trajectories that depart from these boundaries (for backward integration) are bounded sets. Moreover, we also show that the unstable manifolds are geometrically similar to the existing attracting sets. In fact, when no homo- (hetero-)clinic orbits exist, the attractors are omega-limit sets of initial conditions on the unstable manifolds. (C) 2003 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)133-147
Number of pages15
JournalPhysica. D, Nonlinear Phenomena
Issue number3-4
Early online date27 Oct 2003
Publication statusPublished - 15 Dec 2003


  • homoclinic orbits
  • bifurcation
  • nonlinear piecewise systems
  • numerical computation
  • Lorentz
  • model


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