Scaling behaviour of non-hyperbolic coupled map lattices

Details Scaling behaviour of non-hyperbolic coupled map lattices Scaling behaviour of non-hyperbolic coupled map lattices

2006-03-29

arxiv.org/abs/nlin/0603067v2

Phys. Rev. E74 (2006) 046216

Nonlinear Sciences

Chaotic Dynamics

5 pages, including 5 encapsulated PostScript figures

Scientific paper

10.1103/PhysRevE.74.046216

Coupled map lattices of non-hyperbolic local maps arise naturally in many physical situations described by discretised reaction diffusion equations or discretised scalar field theories. As a prototype for these types of lattice dynamical systems we study diffusively coupled Tchebyscheff maps of N-th order which exhibit strongest possible chaotic behaviour for small coupling constants a. We prove that the expectations of arbitrary observables scale with \sqrt{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 \log N^2 modulating the \sqrt{a}-dependence of a given expectation value. We develop a general 1st order perturbation theory to analytically calculate the invariant 1-point density, show that the density exhibits log-periodic oscillations in phase space, and obtain excellent agreement with numerical results.

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