The continuous route to multi-chaos

TL;DR

This paper investigates how high-dimensional chaotic attractors transition from mono-chaos to multi-chaos through a continuous bifurcation, revealing a universal route involving periodic orbit bifurcations.

Contribution

It introduces the concept of multi-chaos, explores its emergence via a continuous bifurcation, and provides small-scale examples to understand high-dimensional chaotic phenomena.

Findings

Identifies a universal route from mono-chaos to multi-chaos.

Shows multi-chaos bifurcation involves creation of periodic orbits with different unstable dimensions.

Provides paradigmatic examples illustrating the transition mechanism.

Abstract

For low-dimensional chaotic attractors there is usually a single number of unstable dimensions for all of its periodic orbits and we can say such attractors exhibit "mono-chaos". In high-dimensional chaotic attractors, trajectories are prone to travel through quite different regions of phase space, some far more unstable than others. This heterogeneity makes predictability even more difficult than in low-dimensional homogeneous chaotic attractors. A chaotic attractor is "multi-chaotic" if every point of the attractor is arbitrarily close to periodic points with different numbers of unstable dimensions. We believe that most physical systems possessing a high-dimensional attractor are of this type. We make three conjectures about multi-chaos which we explore using three two-dimensional paradigmatic examples of multi-chaotic attractors. They can be thought of as small-scale examples that give insight for real high-dimensional phenomena. We find a single route from mono-chaos to multi-chaos if an attractor changes continuously as a parameter is varied. This multi-chaos bifurcation (MCB) is a periodic orbit bifurcation; one branch of periodic orbits is created with a number of unstable dimensions that is different from the mono-chaos.