Clustering of exponentially separating trajectories

TL;DR

This paper investigates the surprising phenomenon where trajectories in certain dynamical systems cluster together despite exponential divergence, demonstrating this effect in a random map but not in the logistic map.

Contribution

It introduces the concept of trajectory clustering with power-law density divergence and demonstrates this effect in a one-dimensional random map, contrasting it with deterministic maps.

Findings

Clustering can occur with power-law divergence in random maps.

No evidence of this clustering in the chaotic logistic map.

The effect is less observable in deterministic systems.

Abstract

It might be expected that trajectories for a dynamical system which has no negative Lyapunov exponent (implying exponential growth of small separations will not cluster together. However, clustering can occur such that the density $\rho(\Delta x)$ of trajectories within distance $\Delta x$ of a reference trajectory has a power-law divergence, so that $\rho(\Delta x)\sim \Delta x^{-\beta}$ when $\Delta x$ is sufficiently small, for some $0<\beta<1$. We demonstrate this effect using a random map in one dimension. We find no evidence for this effect in the chaotic logistic map, and argue that the effect is harder to observe in deterministic maps.