Scaling behavior of nonhyperbolic coupled map lattices

Abstract

Coupled map lattices of nonhyperbolic local maps arise naturally in many physical situations described by discretized reaction diffusion equations or discretized scalar field theories. As a prototype for these types of lattice dynamical systems we study diffusively coupled Tchebyscheff maps of Nth order which exhibit strongest possible chaotic behavior for small coupling constants a. We prove that the expectations of arbitrary observables scale with sqrt of a in the low-coupling limit, contrasting the hyperbolic case which is known to scale with a . Moreover we prove that there are log-periodic oscillations of period ln N2 modulating the sqrt of a dependence of a given expectation value. We develop a general 1st order perturbation theory to analytically calculate the invariant one-point density, show that the density exhibits log-periodic oscillations in phase space, and obtain excellent agreement with numerical results.