Multi-chaos from Quasiperiodicity

TL;DR

This paper introduces the concept of multi-chaos in dynamical systems, characterized by dense trajectories and multiple dense sets of unstable periodic points with different dimensions, using a 2D quasiperiodic orbit as a key example.

Contribution

It defines multi-chaos, a new property in high-dimensional chaotic systems, and presents a simple 2D model demonstrating this behavior with quasiperiodic orbits instead of hyperbolic sets.

Findings

Multi-chaos characterized by dense trajectories and multiple unstable periodic points.

A 2D quasiperiodic orbit can induce multi-chaos, replacing hyperbolic sets.

The concept broadens understanding of chaos in high-dimensional systems.

Abstract

One of the common characteristics of chaotic maps or flows in high dimensions is "unstable dimensional variability", in which there are periodic points whose unstable manifolds have different dimensions. In this paper, in trying to characterize such systems we define a property called "multi-chaos". A set $X$ is multi-chaotic if $X$ has a dense trajectory and for at least 2 values of $k$, the $k$-dimensionally unstable periodic points are dense in $X$. All proofs that such a behavior holds have been based on hyperbolicity in the sense that (i) there is a chaotic set $X$ with a dense trajectory and (ii) in X there are two or more hyperbolic sets with different unstable dimensions. We present a simple 2-dimensional paradigm for multi-chaos in which a quasiperiodic orbit plays the key role, replacing the large hyperbolic set.