Clustering of exponentially separating trajectories

Wilkinson, M.; Mehlig, B.; Gustavsson, K. and Werner, E. (2012). Clustering of exponentially separating trajectories. European Physical Journal B - Condensed Matter and Complex Systems, 85 pp. 18–22.

DOI: https://doi.org/10.1140/epjb/e2011-20325-5

Abstract

It might be expected that trajectories of 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 ρ(Δx) of trajectories within distance |Δx| of a reference trajectory has a power-law divergence, so that ρ(Δx)~|Δx|^−β when |Δx| is sufficiently small, for some 0 < β < 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.

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