Experimentally Accessible Orbits Near a Bykov Cycle

Abstract

This paper reports numerical experiments done on a two-parameter family of vector fields which unfold an attracting heteroclinic cycle linking two saddle-foci. We investigated both local and global bifurcations due to symmetry breaking in order to detect either hyperbolic or chaotic dynamics. Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is still out of reach, using a combination of theoretical tools and computer simulations we have uncovered some complex patterns. We have selected suitable initial conditions to analyze the bifurcation diagrams, and regarding these solutions we have located: (a) an open domain of parameters with regular dynamics; (b) infinitely many parabolic-type curves associated to homoclinic Shilnikov cycles which act as organizing centers; (c) a crisis region related to the destruction or creation of chaotic attractors; (d) a large Lebesgue measure set of parameters where chaotic regimes are dominant, though sinks and chaotic attractors may coexist, and in whose complement we observe shrimps.

Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $

“…We therefore believe that chaotic behavior may still exist in these areas; however, it seems practically unaccessible in an experimental setup. The interplay between regions with accessible and regions with hard to find chaotic dynamics associated with (bifurcations arising from) a Bykov T-point has also been reported in [5].…”

“…The second factor of ( 12) contains the information about the stability in the directions transverse to Fix(η), and its roots with I given by (6) correspond to pitchfork bifurcations; this provides an implicit equation for the locus P. Its parameterization in (iii) is obtained by using instead I = r 2 as obtained from (5) in the second factor of (12), and subsequently solving for f (φ) and δ(φ). Note that the parametrisation in ( 9) is valid for any κ ∈ R and that the range of φ follows from 4κ − 2 tan(φ) > 0, that is, the requirement that the quantity inside the square root be positive.…”