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A model of many symmetrically and locally coupled chaotic oscillators is studied. Partial chaotic synchronizations associated with spontaneous spatial ordering are demonstrated. Very rich patterns of the system are revealed, based on partial synchronization analysis. The stabilities of different partially synchronous spatiotemporal structures and some dynamical behaviors of these states are discussed both numerically and analytically.

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The mean Beld in a globally coupled system of chaotic logistic maps does not obey the standard rules of statistics, even for systems of very large sizes. This indicates the existence of intrinsic instabilities in its evolution. Here these instabilities are related to the very nonsmooth behavior of mean values in a single logistic map, as a function of its parameter. Problems of this kind do not affect a similar system of coupled tent maps, where good statistical behavior has been found. We also explore the transition between these two regimes.PACS number(s): 05.45.+b, 05.90.+m

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

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