The numerical search for the internal dynamics of NHIMs and their pictorial representation

The topic of this article is the numerical search of codimension 2 Normally Hyperbolic Invariant Manifolds (NHIM) in Hamiltonian systems with 3-degrees of freedom and their internal dynamics. We point out relations between different strategies to find such surfaces numerically. We can start from ind...

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Bibliographic Details
Published in:Physica. D 2022-08, Vol.436, p.133330, Article 133330
Main Authors: Gonzalez Montoya, Francisco, Jung, Christof
Format: Article
Language:English
Subjects:
Chaotic scattering
Control chaos
Normally Hyperbolic Invariant Manifolds
Phase space visualization
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Summary:The topic of this article is the numerical search of codimension 2 Normally Hyperbolic Invariant Manifolds (NHIM) in Hamiltonian systems with 3-degrees of freedom and their internal dynamics. We point out relations between different strategies to find such surfaces numerically. We can start from index-1 saddles of the effective potential or from a partially integrable case and follow the NHIM along some curve in the parameter space of the system. Or, we can look for the stable and unstable manifolds of such surfaces by an appropriate indicator method. We show numerical examples for an electron moving in a perturbed magnetic dipole field. •Visualization of invariant manifolds of NHIMs for three degrees of freedom systems.•Detection of bifurcations of NHIMs using the delay time as an indicator.•Method to calculate internal dynamics of NHIMs using a variant of control of chaos.•Bifurcation scenario of NHIMs for Hamiltonian systems with three degrees of freedom.
ISSN:0167-2789
1872-8022
DOI:10.1016/j.physd.2022.133330