Global dynamics in the Poincaré ball of the chen system having invariant algebraic surfaces

Jaume Llibre, Marcelo Messias, Paulo Ricardo Da Silva

Research output: Contribution to journalArticleResearchpeer-review

24 Citations (Scopus)

Abstract

In this paper, we perform a global analysis of the dynamics of the Chen system ẋ= a(y-x), ẏ =(c-a)x-xz+cy, ż =xy-bz, where (x, y, z) ∈ ℝ3 and (a, b, c) ∈ ℝ3. We give the complete description of its dynamics on the sphere at infinity. For six sets of the parameter values, the system has invariant algebraic surfaces. In these cases, we provide the global phase portrait of the Chen system and give a complete description of the α- and ω-limit sets of its orbits in the Poincaré ball, including its boundary S2, i.e. in the compactification of ℝ3 with the sphere S2 of infinity. Moreover, combining the analytical results obtained with an accurate numerical analysis, we prove the existence of a family with infinitely many heteroclinic orbits contained on invariant cylinders when the Chen system has a line of singularities and a first integral, which indicates the complicated dynamical behavior of the Chen system solutions even in the absence of chaotic dynamics. © 2012 World Scientific Publishing Company.
Original languageEnglish
Article number1250154
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume22
Issue number6
DOIs
Publication statusPublished - 1 Jan 2012

Keywords

  • Chen system
  • Poincaré compactification
  • dynamics at infinity
  • heteroclinic orbits
  • integrability
  • invariant manifolds
  • singularly degenerate heteroclinic cycles

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