Clustering of exponentially separating trajectories arXiv:1001.2788v1 [nlin.CD] 16 Jan 2010 M. Wilkinson $^{(1)}$, B. Mehlig$^{(2)}$, K. Gustavsson$^{(2)}$, E. Werner(2) $^{(1)}$Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England $^{(2)}$Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden 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 ρ (Δ x ) of trajectories within distance Δ x of a reference trajectory has a power-law divergence, so that ρ (Δ x ) ∼ Δ x $_{−}$β when Δ x 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. PACS numbers: 05.40.-a,05.45-a 1. Introduction . It is well known that the trajecto- ries of simple dynamical systems can approach fractal sets known as strange attractors [1, 2]. These arise when the largest Lyapunov exponent is positive, implying that trajectories diverge exponentially, but where there is con- traction (characterised by negative Lyapunov exponents) in other directions, so that volume elements contract. Here we demonstrate an alternative mechanism for tra- jectories of a dynamical system to exhibit fractal clus- tering effects. We demonstrate, explain and analyse how fractal clustering of trajectories may occur in a system which has no negative Lyapunov exponents, so that there is no attractor. Fractal measures which arise in the study of dynami- cal systems are typically multifractal, in the sense that their Renyi dimensions D$_{q}$ (discussed in [2, 3]) are not all equal. Exponential separation in all directions implies that the invariant density must cover a set of finite mea- sure, so that for systems with only positive Lyapunov exponents the box-counting dimension D$_{0}$ is equal to the space dimension, d . Furthermore, the Kaplan-Yorke for- mula [4] for the Lyapunov dimension D$_{L}$ gives D$_{L}$ = d if the sum of the Lyapunov exponents is positive. It is be- lieved that D$_{L}$ = D$_{1}$ [5], so that we conclude that D$_{1}$ = d for the systems which we consider. There does not ap- pear to be any constraint which implies D$_{2}$ = d when all of the Lyapunov exponents are positive. It is, however, hard to conceive of how fractal clustering can occur in a situation where trajectories are separating exponentially in each direction. If a cluster with arbitrarily high den- sity is to form, the trajectories which participate have to ‘beat the odds’ by coming closer together throughout a long sequence of iterations of the map, despite the fact that positive Lyapunov exponents imply that separations are more likely to increase than to decrease. 2. The Lyapunov exponent and correlation dimension . We will discuss the simplest case which illustrates our conclusion, which is that of chaotic maps in one dimen- sion. The extensions to flows and to higher dimensions are straightforward. We start by reviewing the defini- tions of the Lyapunov exponent λ and of the correlation dimension D$_{2}$ . It is possible to describe the fractal at- tractor of a deterministic map by considering a measure on the set of points visited by a single trajectory. In this paper, however, we shall consider random as well as deterministic maps, so that the fractal measures upon which trajectories might congregate are not fixed in the coordinate space. We will, therefore, describe clustering in terms of the behaviour of trajectories with different initial conditions, examined at the same ‘time’ (that is, iteration number). The initial distribution of the trajec- tories is a random scatter across the coordinate space, with uniform density. The infinitesimal separation δx of two trajectories is characterised by the Lyapunov exponent, λ [2]. For a one-dimensional map generating a sequence of points x$_{1}$, x$_{2}$, . . . , x$_{i}$, . . . this is defined by writing λ = lim N →∞ 1 N 〈 ln δx$_{N}$ δx$_{1}$ 〉 (1) where the δx$_{i}$ are infinitesimal separations of two tra- jectories at the i th iteration (throughout this paper 〈 X 〉 will denote the expectation value of X ). If λ < 0, the separation of two very close orbits will approach zero with a probability which approaches unity as the initial separation approaches zero. If λ > 0, the separation of two orbits which are initially very close increases with a probability which approaches one as the initial separation decreases. Given this observation, it would be surprising to see trajectories clustering together if λ > 0. Here we will show that clustering may occur, such that the proba- bility density ρ for the separation of trajectories, Δ x has a power-law dependence for small values of Δ x , i.e. ρ (Δ x ) ∼ Δ x − β (2) D$_{q}$ = 1 q − 1 lim ϵ → 0 ln ∑ $_{i}$p q i ln ϵ . (3) with 0 < β < 1. The Renyi dimensions D$_{q}$ are defined by dividing the configuration space into cells, labelled by an index i , with size ϵ , which are occupied with probability p$_{i}$ : The dimension D$_{2}$ is known as the correlation dimension and is related to the exponent β in (2). It follows from