There are few explicit examples in the literature of vector fields exhibiting observable chaos that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of SO(2) ⊕ Z2–symmetric vector fields whose organising center exhibits an attracting heteroclinic network linking two saddle-foci. Each vector field in the family is the restriction to S3 of a polynomial vector field in R4. We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a mechanism called Torus-Breakdown. We explain how an attracting torus gets destroyed by following the changes in the unstable manifold of a saddle-focus.
Although a complete understanding of the corresponding bifurcation diagram and the mecha- nisms underlying the dynamical changes is out of reach, we uncover complex patterns for the symmetric family under analysis, using a combination of theoretical tools and computer sim- ulations. This article suggests a route to obtain rotational horseshoes and strange attractors; additionally, we make an attempt to elucidate some of the bifurcations involved in an Arnold tongue.
Year of publication: 2021
Date published: 08/2021
Type: Original Article