Horseshoes near homoclinic orbits for piecewise linear differential systems in ℝ3

Jaume Llibre, Enrique Ponce, Antonio E. Teruel

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Abstract

For a three-parametric family of continuous piecewise linear differential systems introduced by Arneodo et al. [1981] and considering a situation which is reminiscent of the Hopf-Zero bifurcation, an analytical proof on the existence of a two-parametric family of homoclinic orbits is provided. These homoclinic orbits exist both under Shil'nikov (0 < δ < 1) and non-Shil'nikov assumptions (δ ≥ 1). As it is well known for the case of differentiable systems, under Shil'nikov assumptions there exist infinitely many periodic orbits accumulating to the homoclinic loop. We also prove that this behavior persists at δ= 1. Moreover, for δ > 1 and sufficiently close to 1 we show that these periodic orbits persist but then they do not accumulate to the homoclinic orbit. © World Scientific Publishing Company.
Original languageEnglish
Pages (from-to)1171-1184
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume17
DOIs
Publication statusPublished - 1 Jan 2007

Keywords

  • Homoclinic orbits
  • Horseshoes
  • Piecewise linear differential systems

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    Llibre, J., Ponce, E., & Teruel, A. E. (2007). Horseshoes near homoclinic orbits for piecewise linear differential systems in ℝ3. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 17, 1171-1184. https://doi.org/10.1142/S0218127407017756