Existence of invariant tori in three dimensional maps with degeneracy

TL;DR

This paper proves a KAM-type theorem demonstrating the persistence of two-dimensional invariant tori in degenerate three-dimensional maps, with implications for fluid flow transport barriers, supported by simulations.

Contribution

It establishes a novel KAM result for degenerate action-angle-angle maps with fewer action variables, extending invariant tori persistence theory.

Findings

Invariant tori persist under perturbations in degenerate maps.

Simulation confirms existence of tori in fluid flow models.

Results suggest barriers to transport in 3D fluid flows.

Abstract

We prove a KAM-type result for the persistence of two-dimensional invariant tori in perturbations of integrable action-angle-angle maps with degeneracy, satisfying the intersection property. Such degenerate action-angle-angle maps arise upon generic perturbation of three-dimensional volume-preserving vector fields, which are invariant under volume-preserving action of $S^1$ when there is no motion in the group action direction for the unperturbed map. This situation is analogous to degeneracy in Hamiltonian systems. The degenerate nature of the map and the unequal number of action and angle variables make the persistence proof non-standard. The persistence of the invariant tori as predicted by our result has implications for the existence of barriers to transport in three-dimensional incompressible fluid flows. Simulation results indicating existence of two-dimensional tori in a perturbation of swirling Hill's spherical vortex flow are presented.