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

TL;DR

This paper extends converse KAM theory to provide conditions for the nonexistence of invariant tori transverse to certain foliations, with applications to 3D volume-preserving flows like particle dynamics and fluid flows.

Contribution

It develops a new theoretical framework for proving the nonexistence of invariant surfaces in dynamical systems using converse KAM conditions and numerical implementations.

Findings

Numerical verification for particle in two-wave potential.

Application to Beltrami flows demonstrating theory.

Guidelines for selecting foliations to detect invariant tori.

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.