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arXiv · 2505.20572

History-Dependent Dynamical Invariants in the Lorenz System

B. A. Toledo
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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.

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B. A. Toledo. 2025-05-26. History-Dependent Dynamical Invariants in the Lorenz System. https://doi.org/10.1016/j.chaos.2025.117468

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