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arXiv · 2010.12116

Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory

Nathan Duignan·James D. Meiss
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Abstract

Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend MacKay's converse KAM condition to obtain a sufficient condition for the nonexistence of invariant surfaces that are transverse to a chosen 1D foliation. We show how useful foliations can be constructed from approximate integrals of the system. This theory is implemented numerically for two models, a particle in a two-wave potential and a Beltrami flow studied by Zaslavsky (Q-flows). These are both 3D volume-preserving flows, and they exemplify the dynamics seen in time-dependent Hamiltonian systems and incompressible fluids, respectively. Through both numerical and theoretical considerations, it is revealed how to choose foliations that capture the nonexistence of invariant tori with varying homologies.

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BibTeXRIS

Nathan Duignan, James D. Meiss. 2020-10-23. Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory. https://doi.org/10.1063/5.0035175

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