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The Dynamical Core of a Homoclinic Orbit

Abstract

The complexity of a dynamical system exhibiting a homoclinic orbit is given by its dynamical core which, due to Cantwell, Conlon and Fenley, is a set uniquely determined in the isotopy class, up to a topological conjugacy, of the end-periodic map relative to that orbit. In this work we prove that a sufficient condition to determine the dynamical core of a homoclinic orbit of a Smale diffeomorphism on the \(2\)-disk is the non-existence of bigons relative to this orbit. Moreover, we propose a pruning method for eliminating bigons that can be used to find a Smale map without bigons and hence for finding the dynamical core.

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ACKNOWLEDGMENTS

I would like to thank the Institute IME-USP for the hospitality during the time in which part of this work was done.

Funding

This work was supported by the FAPESP grant 2010/20159-6.

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Correspondence to Valentín Mendoza.

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MSC2010

37E30, 37E15, 37C29, 37B10, 37D20

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Mendoza, V. The Dynamical Core of a Homoclinic Orbit. Regul. Chaot. Dyn. 27, 477–491 (2022). https://doi.org/10.1134/S1560354722040062

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  • DOI: https://doi.org/10.1134/S1560354722040062

Keywords

  • Homoclinic orbits
  • dynamical core
  • Smale horseshoe
  • pruning theory