3 have seen that the correlation dimension is given b y the asymptotic expression Thus clustering is expected to occur for a chaotic one- dimensional map upon varying a parameter so tha t the Lyapunov exponent approaches zero (from above). The only way to avoid this conclusion is if the diffusion con- stant D approaches zero at the same time as the Lya- punov exponent λ . PSfrag repl acements 4. Correlated random walk model . We shall no w illus- trate the clustering phenomenon using a simple ra ndom dynamical system, namely a random walk for whic h the random displacement is a smoothly varying funct ion of the current position. The map is and the function g$_{n}$ ( x ) is a realisation of a rando m pro- cess with statistics specified by a function C ( x − x $^{′}$): Furthermore we consider the coordinate x to b e cyclic, so that the positions x and x + L are equivalent. (This implies that the functions g$_{n}$ ( x ) are periodic with period L ). This model has been discussed in earlier work [8] where it was pointed out that the Lyapunov exponent is negative when the functions g$_{n}$ ( x ) are sufficiently small. The model is of some physical interest as a model for the motion of particles advected by a spatially smooth velocity field, such as a fluid flow [8]. Its attraction in the present context is that it allows us to write down and analyse an exact (although implicit) equation for the correlation dimension, D$_{2}$ = α . Because there is no correlation between the random displacements at successive iterations, the probability density ρ$_{n}$$_{+1}$ for Y$_{n}$$_{+1}$ may be expressed exactly in terms of density ρ$_{n}$ of Y$_{n}$ . For sufficiently small Δ x we have: where P ( Z ) is the probability density of Z$_{n}$ = | 1+ g $^{′}$( x$_{n}$ ) | . This equation has a steady-state solution of the form (11), that is ρ$_{s}$ ( Y ) = A exp( αY ) for some constant A and for an appropriate choice of α . By substituting (11) into (16), we find that α satisfies It is a simple exercise to reproduce equation (13) from this equation by writing 〈 exp( − αZ ) 〉 = 1 − α 〈 Z 〉 + 1 $_{2}$α$^{2}$[ 〈 Z $^{2}$〉 − 〈 Z 〉 $^{2}$] + O ( α $^{3}$). If equation (17) has a solu- tion with 0 < α < 1, this demonstrates conclusively that fractal clustering exists for a model with only positive Lyapunov exponents. We made a numerical investigation of the model de- scribed by (14) and (15), in the case where the noise has a Gaussian distribution and where the correlation func- tion is C ( x ) = σ $^{2}$ξ$^{2}$exp( − x $^{2}$/ 2 ξ $^{2}$) (for some constants σ and ξ ). The density of trajectories in this model is illustrated in figure 1, where we show a scatter plot of the positions for a realisation of the dynamics described by (14) and (15) as a function of iteration number, for a case where the Lyapunov exponent is positive. This shows that the trajectories cluster together, rather than becoming uniformly distributed, as might be expected. The probability density ρ (Δ x ) for a trajectory to occur at a distance Δ x from a randomly chosen test trajec- tory was found to have a power-law form, as described by equation (2), implying that the distribution can be regarded as a fractal set, with correlation dimension D$_{2}$ . As well as confirming that clustering occurs, it is inter- esting to consider the quantitative predictions from (17) in the case where the random functions g$_{n}$ ( x ) have Gaus- sian statistics. Specifically, we assume that the quantities G$_{n}$ = g $^{′}$( x$_{n}$ ) are Gaussian distributed, with variance σ 2 (the mean value is obviously zero). In this case the rela- tion (17) becomes where Γ is the Euler gamma function and$_{1}$ F$_{1}$ is the Kum- mer confluent hypergeometric function. In the limit as σ → ∞ the exponent D$_{2}$ = α approaches unity as so that the clustering tendency is always present in (14), (15), no matter how strong the random impulses g$_{n}$ ( x ). The value of α which solves equation (18) is plotted as a function of σ 2 in figure 2, where it is compared with the asymptotic approximations, (13) and (19) and with values of the exponent obtained by simulation. 5. Extension to deterministic maps . One dimensional deterministic maps can be chaotic, with a positive Lya- punov exponent. It is natural to ask whether these can also exhibit clustering behaviour. The best studied ex- ample [2] is the logistic map, x$_{n}$$_{+1}$ = Kx$_{n}$ (1 − x$_{n}$ ) = D$_{2}$ = λ/ D . (13) x$_{n}$$_{+1}$ = x$_{n}$ + g$_{n}$ ( x$_{n}$ ) (14) 〈 g$_{n}$ ( x ) 〉 = 0 , 〈 g$_{n}$ ( x ) g$_{m}$ ( x $^{′}$) 〉 = δ$_{nm}$C ( x − x $^{′}$) . (15) ρ$_{n}$$_{+1}$ ( Y ) = ∫ ∞ −∞ d Z P ( Z ) ρ$_{n}$ ( Y − Z ) (16) 〈 exp( − αZ ) 〉 = ∫ ∞ −∞ d Z exp( − αZ ) P ( Z ) = 1 . (17) 1 = ∫ ∞ −∞ d G δ ( Z − ln | 1 + G | ) P ( G ) exp( − αZ ) = 1 √$_{2}$$_{πσ}$ ∫ ∞ −∞ d G exp( − G $^{2}$/ 2 σ $^{2}$) | 1 + G | − α = 1 √$_{π}$$_{2}$ α/ $^{2}$σ $_{α}$Γ ( 1 − α 2 ) $_{1}$F$_{1}$ [ α 2 ; 1 2 ; − 1 2 σ 2 ] . (18) D$_{2}$ ∼ 1 − σ − 1 (19) FIG. 1: Illustrating clustering of the trajectories of the corre- lated random walk model defined by equations (14) and (15). 0 200 400 600 800 1000 0 0.5 1 n x