Resonant tori, transport barriers, and chaos in a vector field with a Neimark–Sacker bifurcation

•This work presents a detailed numerical study of the dynamics of a three dimensional dissipative vector field exhibiting a Neimark–Sacker bifurcation.•We follow the attracting invariant torus resulting from this bifurca- tion until it’s destruction and the subsequent appearance of a chaotic attract...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2020-06, Vol.85, p.105226, Article 105226
Main Authors: Fleurantin, Emmanuel, James, J.D. Mireles
Format: Article
Language:English
Subjects:
Canonical forms
Fields (mathematics)
Fluid dynamics
Hopf bifurcation
Invariant tori
Magnetic fields
Neimark–Sacker bifurcation
Numerical analysis
Parameterization method
Parameters
Studies
Topological manifolds
Toruses
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Summary:•This work presents a detailed numerical study of the dynamics of a three dimensional dissipative vector field exhibiting a Neimark–Sacker bifurcation.•We follow the attracting invariant torus resulting from this bifurca- tion until it’s destruction and the subsequent appearance of a chaotic attractor.•We provide a reliable method for computation and visualization of a resonant torus, especially in the regime where the torus is only a C0 invariant object.•We study the stable/unstable manifolds of the equilibrium solutions which act as separatrices/transport barriers for the system.•The numerical study results in a number of conjectures about the global dynamics of the system which we formalize in the conclusions. We make a detailed numerical study of a three dimensional dissipative vector field derived from the normal form for a cusp-Hopf bifurcation. The vector field exhibits a Neimark–Sacker bifurcation giving rise to an attracting invariant torus. Our main goals are to (A) follow the torus via parameter continuation from its appearance to its disappearance, studying its dynamics between these events, and to (B) study the embeddings of the stable/unstable manifolds of the hyperbolic equilibrium solutions over this parameter range, focusing on their role as transport barriers and their participation in global bifurcations. Taken together the results highlight the main features of the global dynamics of the system.
ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2020.105226