Title: 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.