Quadratic vector fields equivariant under the D2 symmetry group

Stavros Anastassiou, Spyros Pnevmatikos, Tassos Bountis

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


Symmetry often plays an important role in the formation of complicated structures in the dynamics of vector fields. Here, we study a specific family of systems defined on 3, which are invariant under the D2 symmetry group. Under the assumption that they are polynomial of degree at most two, they belong to a two-parameter family of vector fields, called the D 2 model. We describe the global behavior of the system, for most parameter values, and locate a region of parameter space where complicated structures occur. The existence of heteroclinic and homoclinic orbits is shown, as well as of heteroclinic cycles (for other parameter values), implying the presence of (different types of) Shil'nikov type of chaos in the D2 systems. We then employ Poincaré maps to illustrate the bifurcations leading to this behavior. The global bifurcations exhibited by its strange attractors are explained as an effect of symmetry. We conclude by describing the behavior of the system at infinity.

Original languageEnglish
Article number1350017
JournalInternational Journal of Bifurcation and Chaos
Issue number1
Publication statusPublished - Jan 2013


  • Symmetric systems
  • behavior at infinity
  • global bifurcations

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics

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