2 where the functions f$_{n}$ ( x ) may be independent of n (that is, the map may be deterministic), or they are selected at random from an ensemble at each iteration. We concen- trate upon chaotic maps, where the Lyapunov exponent is positive. We will assume that the iterations do not escape from some finite region, so that there is an upper bound on the separation of two trajectories. We allow for the possibility of a random map because the analysis can be taken further in that case and because random maps play a role in modelling many different physical processes. Both the Lyapunov exponent λ and the clustering ex- ponent β characterise small separations of trajectories, Δ x . As Δ x → 0 these become infinitesimal separations, denoted by δx . The evolution of the infinitesimal sepa- rations is described by the linearised equation of motion, δx$_{n}$$_{+1}$ = f ′ $_{n}$( x$_{n}$ ) δx$_{n}$ . It is convenient to consider the variable Y$_{n}$ = lnΔ x$_{n}$ , which we express in terms of its incremental changes at each iteration, Z$_{n}$ : where, in the limit where Δ x$_{i}$ is sufficiently small, we have Even for a deterministic map, the quantity Z$_{n}$ differs from one iteration to the next, and if the map is chaotic this variation will appear to be random. We now consider how both the Lyapunov exponent λ and the clustering exponent β are related to the statistics of Z . It is clear from the definitions (1) and (5) that Let us consider dynamics of Y$_{n}$ . Because the displace- ments Z$_{n}$ can be considered as random variables, the quantity Y$_{n}$ executes a random walk with drift when Δ x$_{n}$ is sufficiently small. The variable Y has diffusive fluctu- ations, with the variance of Δ Y$_{n}$ = Y$_{i}$$_{+}$$_{n}$ − Y$_{i}$ increasing linearly as a function of the ‘time’, n : The diffusion coefficient D is determined from the corre- lation function of the fluctuations We characterise the behaviour of Y$_{n}$ in terms of its prob- ability density, ρ$_{n}$ ( Y ). If the probability density ρ$_{n}$ ( Y ) varies sufficiently slowly as a a function of Y , we may approximate its evolution by discrete-time version of the Fokker-Planck equation [6]: where the drift velocity v = 〈 Z 〉 is, by (7), equal to the Lyapunov exponent. The steady-state solution of equa- tion (10), ρ$_{s}$ , is an exponential function: (3) that the number N of trajectories in a ball of ra- dius ϵ about a randomly selected test trajectory satisfies 〈N ( ϵ ) 〉 ∼ ϵ D$_{2}$ [2]. The probability density for trajectory separations in one dimension is ρ ( ϵ ) = d 〈N 〉 / d ϵ , so that D$_{2}$ = 1 − β . 3. Analysis of clustering . We consider the dynamics of a one-dimensional map, of the form where α = v/ D . This solution is not normalisable and we should therefore consider the conditions under which it is applicable. The linearised mapping ceases to be ap- plicable when Δ x is too large. Because the growth of Δ x is assumed to be bounded, the probability density ρ$_{s}$ ( Y ) should have a normalisable steady state. We can there- fore meaningfully consider a solution of the form (11) with positive values of α , because the solution is matched to another function at some cutoff value. However there is no lower cutoff, so that solutions with negative values of α are untenable. Now consider the implications of (11) for the probabil- ity density of trajectory separations. Transforming (11) to a density ρ (Δ x ) by writing d P = A exp( αY )d Y = ρ (Δ x )dΔ x , we find ρ (Δ x ) = A Δ x α − $^{1}$. We conclude that clustering should occur when α < 1, and that the corre- lation dimension is the same as the exponent in (11): we have The use of the Fokker-Planck equation is only justi- fied when the gradient of ρ$_{n}$ ( Y ) is sufficiently small. The condition is that ∂ρ$_{n}$/∂Y should be small compared to 1 /δY$_{0}$ , where δY$_{0}$ is the scale over which Y varies dur- ing its correlation time. This condition for the valid- ity of (11) is equivalent to D$_{2}$ ≪ 1. We remark that Grassberger and Procaccia [7] also used a Fokker-Planck equation for the logarithm of trajectory separations in a discussion of the correlation dimension. They consid- ered a strange attractor, rather than a system with only positive Lyapunov exponents. From these considerations we make the following con- clusions. For any chaotic map, we can calculate the mean value and variance of Z$_{n}$ = ln | f ′ $_{n}$( x$_{n}$ ) | . The mean value is equal to the Lyapunov exponent, λ = 〈 Z 〉 , and for a chaotic map this is a positive number. The fluctuations of Y = lnΔ x execute a random walk, with diffusion co- efficient D . When the probability density of Y varies sufficiently slowly, this obeys a Fokker-Planck equation. The stationary solution of this equation is an exponen- tial which in turn implies that the distribution of Δ x is a power-law. The assumption that the probability den- sity is slowly varying is justified when D$_{2}$ = α ≪ 1, a limit in which there is strong clustering. In this limit we x$_{n}$$_{+1}$ = f$_{n}$ ( x$_{n}$ ) (4) ρ$_{n}$$_{+1}$ − ρ$_{n}$ = − ∂ ∂Y ( vρ$_{n}$ ) + ∂ 2 ∂Y $_{2}$( D ρ$_{n}$ ) (10) ρ$_{s}$ ( Y ) = A exp( αY ) (11) Y$_{n}$ = ln(Δ x$_{n}$ ) = ln(Δ x$_{0}$ ) +$^{n}$$^{−}$$^{1}$ ∑ i =0 Z$_{i}$ (5) Z$_{n}$ = ln | f ′ $_{n}$( x$_{n}$ ) | . (6) λ = 〈 Z ( t ) 〉 . (7) lim n →∞ 〈 (Δ Y$_{n}$ − nλ ) $^{2}$〉 n = 2 D . (8) D = 1 2 ∞ ∑ n = −∞ [ 〈 Z$_{n}$Z$_{0}$ 〉 − 〈 Z 〉 $^{2}$] . (9) D$_{2}$ = α = 1 − β . (12)