Labyrinth walks: An elegant chaotic conservative non-Hamiltonian system

TL;DR

The paper introduces 'Labyrinth walks', a chaotic conservative system that is non-Hamiltonian, lacks a symplectic structure, yet exhibits properties like a vector potential and fractional Brownian motion-like behavior.

Contribution

It presents a novel example of a chaotic, conservative, non-Hamiltonian system with unique properties and phase space structure, expanding understanding of dynamical systems.

Findings

Does not admit an autonomous Hamiltonian or symplectic structure.

Exhibits chaotic behavior with unstable stationary points in a 3D grid.

Shows motion similar to fractional Brownian motion despite determinism.

Abstract

In this paper, we show that "Labyrinth walks", the conservative version of "Labyrinth chaos" and member of the Thomas-R\"ossler class of systems, does not admit an autonomous Hamiltonian as a constant function in time, and as a consequence, does not admit a symplectic structure. However, it is conservative, and thus admits a vector potential, being at the same time chaotic. This exceptional set of properties makes "Labyrinth walks" an elegant example of a chaotic, conservative, non-Hamiltonian system, with only unstable stationary points in its phase space, arranged in a 3-dimensional grid. As a consequence, "Labyrinth walks", even though is a deterministic system, it exhibits motion reminiscent of fractional Brownian motion in stochastic systems!