Quadratic vector fields equivariant under the D2 symmetry group

Stavros Anastassiou, Spyros Pnevmatikos, Tassos Bountis

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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
Volume23
Issue number1
DOIs
Publication statusPublished - Jan 2013
Externally publishedYes

Fingerprint

Quadratic field
Symmetry Group
Equivariant
Vector Field
Chaos theory
Orbits
Polynomials
Heteroclinic Cycle
Symmetry
Heteroclinic Orbit
Strange attractor
Global Bifurcation
Homoclinic Orbit
Parameter Space
Two Parameters
Chaos
Bifurcation
Infinity
Polynomial
Invariant

Keywords

  • behavior at infinity
  • global bifurcations
  • Symmetric systems

ASJC Scopus subject areas

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

Cite this

Quadratic vector fields equivariant under the D2 symmetry group. / Anastassiou, Stavros; Pnevmatikos, Spyros; Bountis, Tassos.

In: International Journal of Bifurcation and Chaos, Vol. 23, No. 1, 1350017, 01.2013.

Research output: Contribution to journalArticle

Anastassiou, Stavros ; Pnevmatikos, Spyros ; Bountis, Tassos. / Quadratic vector fields equivariant under the D2 symmetry group. In: International Journal of Bifurcation and Chaos. 2013 ; Vol. 23, No. 1.
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