content_image/380924.jpg content_image/380923.jpg
<img src='content_image/380924.jpg'>

f ( x$_{n}$ ) which has been studied exhaustively and which is considered to have generic properties. There are intervals of the parameter K where the trajectory is attracted to stable periodic orbits, interspersed with sets where the Lyapunov exponent is positive. When the correlation di- mension is small it is well approximated by D$_{2}$ = λ/ D . We therefore expect that clustering of trajectories will oc- cur close to the boundaries of the chaotic regions, where λ is small. 

A numerical investigation of the correlation dimension for the logistic map was carried out, for a large set of values of the coupling parameter K . This set included many values close to the edge of a stable region, where the Lyapunov exponent, although positive, was very small. This investigation produced a surprising result. With one exception, no values of K were identified for which the numerically determined correlation dimension was clearly different from unity (in the chaotic zones) or zero (in the stable zones). The only exception was when K was set equal to the value which corresponds to the limit point of the Feigenbaum period-doubling sequence. At that value of K , the results suggested a value of D$_{2}$ which is less than unity but greater than zero, consistent with results by Grassberger [9]. 

[1] J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. , 57 , 617656, (1985). 

[2] E. Ott, Chaos in Dynamical Systems , 2nd edition, Cam- bridge: University Press, (2002). 

[3] H. G. E. Hentschel and I. Procaccia, Physica D , 8 , 435, (1983). 

[4] J. L. Kaplan and J. A. Yorke, in Functional Differential Equations and Approximations of Fixed Points , Lecture Notes in Mathematics, eds. H.-O. Peitgen and H.-O. Wal- ter, Springer, Berlin, 730 , 204, (1979). 

The observation that D$_{2}$ does not become small when λ is small is consistent with (13) if the diffusion con- stant D approaches zero at the same time as λ approaches zero. In the case of deterministic systems, this is in fact what happens, as the following heuristic argument shows. Consider what happens as the parameter K of the logis- tic map is varied from K − , where λ is slightly negative, to the nearby value K$_{+}$ where λ is slightly positive. At K − the map has an attractor which is a stable periodic orbit of period M (say). The Lyapunov exponent for the stable orbit is 

where x ∗ j are the points on the periodic orbit. In the case of the periodic orbit the shift of Y with every orbit is precisely Mλ , implying that D = 0 for the stable orbit. At K$_{+}$ the periodic orbit has either become unstable, or else has ceased to exist. The trajectory then explores a dense subset of the line, but it must spend most of the time in the vicinity of the sequence of M points vis- ited by the nearby stable orbit. While the trajectory is trapped close to the periodic orbit, there is negligible ran- domness in the values of Z . We therefore expect that D approaches zero at the boundary of the unstable region. If a small amount of noise is added to a deterministic map, the value of D remains finite at the boundary of the chaotic region, and we find that D$_{2}$ approaches zero smoothly at the boundary. 

Conclusions . Fractal clustering may occur even if there is no negative Lyapunov exponent. This was anal- ysed and demonstrated in a random dynamical system, namely the correlated random walk. In the case of the logistic map, the clustering effect was not observed. We argued that it is harder to see the clustering effect in a deterministic dynamical system, although there does not appear to be any reason why it is forbidden. It would be interesting to find an example. 

Acknowledgments. The work of BM was supported by Vetenskapsr˚ adet, and that of BM, KG and EW by the platform ‘Nanoparticles in an interactive environment’ at G¨ oteborg university. 

[5] F. Ledrappier and L. S. Young, Comm. Math. Phys. 117 , 529, (1988). 

[6] N. G. van Kampen, Stochastic processes in physics and chemistry , 2nd ed., North-Holland, Amsterdam, (1981). 

[7] P. Grassberger and I. Procaccia, Physica D , 13 , 34-54, (1984). 

[8] M. Wilkinson and B. Mehlig, Phys. Rev. E , 68 , 040101(R), (2003). 

[9] P. Grassberger, J. Stat. Phys. , 26 , 173-9, (1981). 

<img src='content_image/380923.jpg'>

