Partial barriers to chaotic transport in 4D symplectic maps

TL;DR

This paper investigates partial barriers to chaotic transport in 4D symplectic maps, introducing the concept of cantorus-NHIMs and flux formulas to quantify transport restrictions in higher-dimensional Hamiltonian systems.

Contribution

It introduces cantorus-NHIMs as partial barriers in 4D symplectic maps and develops flux formulas to quantify transport across these barriers.

Findings

Defined partial barriers based on cantorus-NHIMs.

Derived a flux formula for 4D partial barriers.

Introduced local 3D flux relevant for Arnold diffusion.

Abstract

Chaotic transport in Hamiltonian systems is often restricted due to the presence of partial barriers, leading to a limited flux between different regions in phase phase. Typically, the most restrictive partial barrier in a 2D symplectic map is based on a cantorus, the Cantor set remnants of a broken 1D torus. For a 4D symplectic map we establish a partial barrier based on what we call a cantorus-NHIM, a normally hyperbolic invariant manifold (NHIM) with the structure of a cantorus. Using a flux formula, we determine the global 4D flux across a partial barrier based on a cantorus-NHIM by approximating it with high-order periodic NHIMs. In addition, we introduce a local 3D flux depending on the position along a resonance channel, which is relevant in the presence of slow Arnold diffusion. Moreover, for a partial barrier composed of stable and unstable manifolds of a NHIM we utilize periodic NHIMs to quantify the corresponding flux.