# High dimensional chaotic systems which behave like random walks in state space

## TL;DR

This paper analyzes an n-dimensional chaotic system that exhibits random walk-like behavior on a hypersphere, revealing how error growth and predictability vary with control parameters and linking it to turbulence.

## Contribution

It introduces a high-dimensional generalization of Thomas's attractor and demonstrates its behavior as a constrained random walk, connecting chaos theory with turbulence predictability.

## Key findings

- Error growth is limited and depends on control parameters.
- Chaotic behavior resembles a constrained random walk.
- Link established between high-dimensional chaos and turbulence predictability.

## Abstract

By analysing an n-dimensional generalisation of Thomas's cyclically symmetric attractor we find that this chaotic dynamical system behaves like a random walk constrained onto the surface of a hypersphere. The growth of error is limited, with qualitatively different behaviour depending on a control parameter. For moderate values of the control parameter, linear growth of error is seen. For low values of the control parameter, the error is limited by the random walk behaviour. Finally, we link this to the predictability of homogeneous isotropic turbulence, which we find here also behaves like a constrained random walk.