History-Dependent Dynamical Invariants in the Lorenz System

TL;DR

This paper uncovers a new class of history-dependent invariants in the Lorenz system, revealing hidden structure and redefining chaos and order coexistence in dissipative systems.

Contribution

It introduces a novel non-local invariant in the Lorenz system, demonstrating its existence and conservation through a constructive phase space augmentation.

Findings

Existence of a non-local invariant in the Lorenz system.

Invariant remains constant along trajectories, including chaotic ones.

Unstable periodic orbits correspond to specific invariant values.

Abstract

Contrary to the established view of the Lorenz system as an archetype of dissipative chaos lacking conserved quantities, this work rigorously demonstrates the existence of a novel class of history-dependent dynamical invariants. Through a constructive method that augments the phase space, we derive a non-local invariant whose value remains constant along any trajectory. Its history-dependence arises from an integral term that accumulates the orbit's past, thereby ensuring its conservation. The invariant's constancy is verified with high-precision numerical simulations for both periodic and chaotic orbits. This finding reveals a hidden structure within the attractor and affords a new physical interpretation where unstable periodic orbits (UPOs) correspond to specific values of this conserved quantity. The result redefines the notion of non-integrability in dissipative systems, showing that non-local order can coexist with chaotic behavior.