Labyrinth Chaos is not Hamiltonian but still has a Vector Potential

TL;DR

This paper proves that Labyrinth chaos systems, despite being volume-preserving and chaotic, do not have a Hamiltonian structure but do admit a vector potential, marking a novel finding in dynamical systems theory.

Contribution

It provides the first proof that a conservative chaotic system can lack a Hamiltonian but still possess a vector potential.

Findings

Labyrinth chaos systems are not Hamiltonian.

They are volume-preserving and chaotic.

A vector potential exists for these systems.

Abstract

We provide here a comprehensive proof that the so-called Labyrinth chaos systems, a member of the Thomas-R\"ossler (TR) class of systems do not admit a Hamiltonian; yet they admit a vector potential. The proof starts from the general case of TR systems, which are in general non-conservative and we show that this is also true for the conservative (volume-preserving) case known as `Labyrinth chaos'. To our knowledge, this is the first instance reported where a conservative chaotic system does not, in principle, admit a Hamiltonian symplectic structure. Still, a vector potential is readily admissible and thus, constructed.