Eur. Phys. J. B (2012) 85: 18
https://doi.org/10.1140/epjb/e2011-20325-5

Regular Article

Clustering of exponentially separating trajectories

¹ Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England
² Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden

^a e-mail: m.wilkinson@open.ac.uk
^b e-mail: Bernhard.Mehlig@physics.gu.se

Received: 26 April 2011
Received in final form: 18 October 2011
Published online: 18 January 2012

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.