Conditions for global synchronization in lattices of chaotic elements withlocal connections

We investigate the phenomena on the edge of spatially homogeneous chaotic m ode and spatiotemporal chaos in a lattice of chaotic 1D maps with local con nections. We show that spatially homogeneous chaotic mode cannot exist in a lattice with local connections if the Lyapunov exponent lambda of the isol ated chaotic map is greater than some critical positive value. We propose a few schemes that make spatial synchronization possible in large lattices. If the idea of only local connections is abandoned, the number of connectio ns necessary for synchronization dramatically decreases to three per node. We also propose a model of a lattice with an external pacemaker, where we f ind a spatially homogeneous mode synchronous with the pacemaker, as well as different from the pacemaker mode.