Analytical results for coupled map lattices with long-range interactions
We obtain exact analytical results for lattices of maps with couplings that decay with distance as $r^{-α}$. We analyze the effect of the coupling range on the system dynamics through the Lyapunov spectrum. For lattices whose elements are piecewise linear maps, we get an algebraic expression for the Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the Lyapunov spectrum for a completely synchronized state is analytically obtained. The critical lines characterizing the synchronization transition are determined from the expression for the largest transversal Lyapunov exponent. In particular, it is shown that in the thermodynamical limit, such transition is only possible for sufficiently long-range interactions, namely, for $α\le alpha_c<d$, where $d$ is the lattice dimension.