Nongeneric bifurcations near heterodimensional cycles with inclination flip in ℝ 4

Dan Liu, Shigui Ruan, Deming Zhu

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


Nongeneric bifurcation analysis near rough heterodimensional cycles associated to two saddles in ℝ 4 is presented under inclination flip. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, we construct a Poincarè return map under the nongeneric conditions and further obtain the bifurcation equations. Coexistence of a heterodimensional cycle and a unique periodic orbit is proved after perturbations. New features produced by the inclination flip that heterodimensional cycles and homoclinic orbits coexist on the same bifurcation surface are shown. It is also conjectured that homoclinic orbits associated to different equilibria coexist.

Original languageEnglish (US)
Pages (from-to)1511-1532
Number of pages22
JournalDiscrete and Continuous Dynamical Systems - Series S
Issue number6
StatePublished - Dec 2011


  • Bifurcation
  • Dichotomy
  • Heterodimensional cycle
  • Homolinic orbit
  • Inclination flip
  • Poincarè map

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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