# Chaos on a High-Dimensional Torus

## TL;DR

This paper investigates the transition from high-dimensional quasiperiodic tori to chaos using coupled circle maps, revealing the existence of a novel form of chaos called toric chaos with unique dynamical properties.

## Contribution

It confirms the existence of high-dimensional tori, introduces the concept of toric chaos, and explores its properties and relevance to neural dynamics and turbulence.

## Key findings

- High-dimensional tori become exponentially rare as dimension increases
- Chaos can occur in invertible maps with multiple null Lyapunov exponents
- Toric chaos exhibits delocalization and slow Lyapunov vector dynamics

## Abstract

Transition from quasiperiodicity with many frequencies (i.e., a high-dimensional torus) to chaos is studied by using $N$-dimensional globally coupled circle maps. First, the existence of $N$-dimensional tori with $N\geq 2$ is confirmed while they become exponentially rare with $N$. Besides, chaos exists even when the map is invertible, and such chaos has more null Lyapunov exponents as $N$ increases. This unusual form of "chaos on a torus," termed toric chaos, exhibits delocalization and slow dynamics of the first Lyapunov vector. Fractalization of tori at the transition to chaos is also suggested. The relevance of toric chaos to neural dynamics and turbulence is discussed in relation to chaotic itinerancy.