Scaling properties of invariant densities of coupled Tchebyscheff maps

Abstract

We study 1-dimensional coupled map lattices consisting of diffusively coupled Tchebyscheff maps of N-th order. For small coupling constants a we determine the invariant 1-point and 2-point densities of these nonhyperbolic systems in a perturbative way. For arbitrarily small couplings a>0 the densities exhibit a selfsimilar cascade of patterns, which we analyse in detail. We prove that there are log-periodic oscillations of the density both in phase space as well as in parameter space. We show that expectations of arbitrary observables scale with \sqrt{a} in the low-coupling limit, contrasting the case of hyperbolic maps where one has scaling with a. Moreover we prove that there are log-periodic oscillations of period \log N^2 modulating the \sqrt{a}-dependence of the expectation value of any given observable.

arXiv comment31 pages, including 19 encapsulated PostScript figures

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• Chaotic strings and standard model parameters
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Cited by in corpus (5)

• Short chaotic strings and their behaviour in the scaling region
• Stochastic differential equations driven by deterministic chaotic maps: analytic solutions of the Perron-Frobenius equation
• Renormalization group approach to chaotic strings
• Statistical mechanics of the vacuum
• A first approach to the Galois group of chaotic chains