In certain physical situations, extensive interactions arise naturally in systems. We consider one such situation, namely, small-world couplings. We show that, for a fixed fraction of nonlocal couplings, synchronous chaos is always a stable attractor in the thermodynamic limit. We point out that randomness helps synchronization. We also show that there is a size dependent bifurcation in the collective behavior in such systems.

In this paper we study the existing observation in literature about synchronization of a large number of coupled maps with random nonlocal connectivity ͓Chate and Manneville, Chaos 2, 307 ͑1992͔͒. These connectivities which lack any spatial significance can be realized in neural nets and electrical circuits. It is quite interesting and of practical importance to note that a huge number of maps can be synchronized with this connectivity. We show that this synchronization stems from the fact that the connectivity matrix has a finite gap in the eigenvalue spectrum in the macroscopic limit. We give a quantitative explanation for the gap. We compare the analytic results with the ones quoted in the above reference. We also study the departures from this highly collective behavior in the low connectivity limit and show that the behavior is almost statistical for very low connectivity.

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We study coupled maps on a Cayley tree, with local (nearest-neighbor) interactions, and with a variety of boundary conditions. The homogeneous state (where every lattice site has the same value) and the node-synchronized state (where sites of a given generation have the same value) are both shown to occur for particular values of the parameters and coupling constants. We study the stability of these states and their domains of attraction. As the number of sites that become synchronized is much higher compared to that on a regular lattice, control is easier to effect. A general procedure is given to deduce the eigenvalue spectrum for these states. Perturbations of the synchronized state lead to different spatio-temporal structures. We find that a mean-field like treatment is valid on this (effectively infinite dimensional) lattice.PACS number(s): 05.45. +b, 47.20. Ky

We study the transition to phase synchronization in a model for the spread of infection defined in a small world network. It was shown [Phys. Rev. Lett. 86, 2909 (2001)] that the transition occurs at a finite degree of disorder p, unlike equilibrium models where systems behave as random networks even at infinitesimal p in the infinite-size limit. We examine this system under variation of a parameter determining the driving rate and show that the transition point decreases as we drive the system more slowly. Thus it appears that the transition moves to p=0 in the very slow driving limit, just as in the equilibrium case.