AI Article Synopsis
- The study presents analytical results for maps on lattices with couplings that decrease with distance following a specific power law (r^(-alpha)).
- The effects of coupling range on the system's behavior are investigated using the Lyapunov spectrum, providing a formula for piecewise linear maps and a computation for synchronized states in nonlinear maps.
- It is concluded that synchronization transitions can occur only with long-range interactions when the coupling decay parameter is less than a critical value, indicated as alpha < alpha(c) = d, where d represents the dimension of the lattice.
Article 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 line characterizing the synchronization transition is 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
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http://dx.doi.org/10.1103/PhysRevE.68.045202 DOI Listing Publication Analysis
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