Action-gradient-minimizing pseudo-orbits and almost-invariant tori

TL;DR

This paper introduces new methods for defining and visualizing almost-invariant tori in near-integrable Hamiltonian systems using action-gradient-minimizing pseudo-orbits, enhancing understanding of transport barriers.

Contribution

It presents three new definitions of pseudo-orbits based on Hamilton's Principle, along with equivalent formulations and a novel visualization approach for action-minimizing orbits.

Findings

Defined almost-invariant tori using pseudo-orbits with different extremization strategies.

Established equivalence between Lagrangian and Hamiltonian formulations.

Proposed a new visualization method for action-minimizing orbits.

Abstract

Transport in near-integrable, but partially chaotic, $1 1/2$ degree-of-freedom Hamiltonian systems is blocked by invariant tori and is reduced at \emph{almost}-invariant tori, both associated with the invariant tori of a neighboring integrable system. "Almost invariant" tori with rational rotation number can be defined using continuous families of periodic \emph{pseudo-orbits} to foliate the surfaces, while irrational-rotation-number tori can be defined by nesting with sequences of such rational tori. Three definitions of "pseudo-orbit," \emph{action-gradient--minimizing} (AGMin), \emph{quadratic-flux-minimizing} (QFMin) and \emph{ghost} orbits, based on variants of Hamilton's Principle, use different strategies to extremize the action as closely as possible. Equivalent Lagrangian (configuration-space action) and Hamiltonian (phase-space action) formulations, and a new approach to visualizing action-minimizing and minimax orbits based on AGMin pseudo-orbits, are presented.