Dynamics of Diffusion-coupled Logistic Maps
by
Harold M. Hastings
Hofstra University

Logistic maps have been widely studied as a model of ßmall-scale, dissipative" chaos. A (two-dimensional) lattice of logistic maps is thus a natural model for spatio-temporal chaos. We shall survey the effects of coupling these logistic maps by a simple model of diffusion. As might be expected, small values of diffusion smooth out the bulk dynamics represented by the average of the state variables. However, further increases in diffusion yield interesting collective behaviors that can be interpreted as a failure of the law of large numbers (Kaneko), a Hopf bifurcation, or a (dynamical) second-order phase transition (Strassinopolous and Alstrom). In addition, we shall see that interesting and in some cases remarkably persistent spatial patterns emerge at large values of diffusion.

Date received: July 24, 1999