Chaos or Order?

Abstract

What is chaos? Despite several decades of research on this ubiquitous and fundamental phenomenon there is yet no agreed-upon answer to this question. Recently, it was realized that all stochastic and deterministic differential equations, describing all natural and engineered dynamical systems, possess a topological supersymmetry. It was then suggested that its spontaneous breakdown could be interpreted as the stochastic generalization of deterministic chaos. This conclusion stems from the fact that such phenomenon encompasses features that are traditionally associated with chaotic dynamics such as non-integrability, positive topological entropy, sensitivity to initial conditions, and the Poincare-Bendixson theorem. Here, we strengthen and complete this picture by showing that the hallmarks of set-theoretic chaos -- topological transitivity/mixing and dense periodic orbits -- can also be attributed to the spontaneous breakdown of topological supersymmetry. We also demonstrate that these features, which highlight the noisy character of chaotic dynamics, do not actually admit a stochastic generalization. We therefore conclude that spontaneous topological symmetry breaking can be considered as the most general definition of continuous-time dynamical chaos. Contrary to the common perception and semantics of the word "chaos", this phenomenon should then be truly interpreted as the low-symmetry, or ordered phase of the dynamical systems that manifest it. Since the long-range order in this case is temporal, we then suggest the word "chronotaxis" as a better representation of this phenomenon.