Duignan Nathan, Meiss James D
Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA.
Chaos. 2021 Jan;31(1):013124. doi: 10.1063/5.0035175.
Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend MacKay's converse Kolmogorov-Arnol'd-Moser 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.
