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Advisor(s)
Abstract(s)
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)⊕Z2𝕊𝕆(2)⊕ℤ2-symmetric vector fields whose organizing center exhibits an attracting heteroclinic network linking two saddle-foci. Each vector field in the family is the restriction to S3𝕊3 of a polynomial vector field in R4ℝ4. 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 mechanisms 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 simulations. 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.
Description
Keywords
Global bifurcation Heteroclinic attractor Arnold tongue Torus-breakdown Strange attractor Symmetry-breaking
Pedagogical Context
Citation
Castro, L., & Rodrigues, A. (2021). Torus-Breakdown Near a Heteroclinic Attractor: A Case Study. International Journal of Bifurcation and Chaos, 31(10), 2130029. https://doi.org/10.1142/s0218127421300299
Publisher
World Scientific