Chaotic Maps, Hamiltonian Flows, and Holographic Methods

TL;DR

This paper introduces holographic functional methods to analyze chaotic maps, revealing their quasi-Hamiltonian structure and explaining chaos phenomena like frequency cascade and trajectory folding through switchback potentials.

Contribution

It presents a novel holographic approach to study discrete chaotic maps, uncovering their underlying quasi-Hamiltonian dynamics and physical interpretation of chaos mechanisms.

Findings

Maps are shown to be quasi-Hamiltonian systems with switchback potentials.

The method explains frequency cascade and trajectory folding in chaos.

Holographic techniques provide new insights into discrete chaotic dynamics.

Abstract

Holographic functional methods are introduced as probes of discrete time-stepped maps that lead to chaotic behavior. The methods provide continuous time interpolation between the time steps, thereby revealing the maps to be quasi-Hamiltonian systems underlain by novel potentials that govern the motion of a perceived point particle. Between turning points, the particle is strictly driven by Hamiltonian dynamics, but at each encounter with a turning point the potential changes abruptly, loosely analogous to the switchbacks on a mountain road. A sequence of successively deepening switchback potentials explains, in physical terms, the frequency cascade and trajectory folding that occur on the particular route to chaos revealed by the logistic map.