Analytical results for coupled map lattices with long-range interactions

Abstract

We obtain exact analytical results for lattices of maps with couplings that decay with distance as $r^{-\alpha }$. 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 $\alpha \le alph{a}_c<d$, where $d$ is the lattice dimension.

4 pages, 2 figures, corrections included. Phys. Rev. E 68, 045202(R) (2003); correction in press

References in corpus (11)

• The sound of many hands clapping
• Long-range interacting rotators: connection with the mean--field approximation
• Long-range interactions and non-extensivity in ferromagnetic spin models
• Canonical solution of a system of long-range interacting rotators on a lattice
• Synchronization of chaotic systems: Transverse stability of trajectories in invariant manifolds
• Synchronization of oscillators with long range interaction: phase transition and anomalous finite size effects
• Coupled Maps on Trees
• Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach
• Chaos suppression in the large size limit for long-range systems
• Transition Temperature of Josephson Junction Arrays with Long-Range Interaction
• Synchronization in large populations of limit cycle oscillators with long-range interactions

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