Low-dimensional paradigms for high-dimensional hetero-chaos

TL;DR

This paper introduces simplified models of hetero-chaotic systems, where different regions of a high-dimensional attractor exhibit varying degrees of instability, providing insights into complex high-dimensional chaotic phenomena.

Contribution

The authors develop simplified models demonstrating hetero-chaos, highlighting the low-dimensional structures underlying high-dimensional chaotic attractors.

Findings

Most physical high-dimensional systems are likely hetero-chaotic.

Simplified models can capture the essential features of hetero-chaos.

Hetero-chaos involves regions with different unstable dimensions.

Abstract

The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some regions than others. Some regions of a chaotic attractor can be expanding in more dimensions than other regions. Imagine a situation where two such regions and each contains trajectories that stay in the region for all time while typical trajectories wander throughout the attractor. Such an attractor is "hetero-chaotic" (i.e. it has heterogeneous chaos) if furthermore arbitrarily close to each point of the attractor there are points on periodic orbits that have different unstable dimensions. This is hard to picture but we believe that most physical systems possessing a high-dimensional attractor are of this type. We have created simplified models with that behavior to give insight to real high-dimensional phenomena.