Scaling in Time Series from Coupled Map Lattices

Authors: Sevgi ÇIKCI, Ayşe ERZAN

Abstract: Coupled Map Lattices(CML)[1] have been introduced to model spatial and temporal complexity in spatially extended systems. We have studied time-series from a single site on a one-dimensional periodic lattice of diffusively coupled logistic maps f(x)=\mu x(1-x), x_i(n+1)=(1-\epsilon)f(x_i(n)) +\epsilon/2(f(x_{i-1}(n)- 2f(x_i(n))+f(x_{i+1}(n)) which we consider as a model for a high-dimensional dynamic system. We find, i) for random initial conditions that x_i(n) obeys a unimodal distribution for small \mu, and exhibits a phase transition to a bimodal distribution at \mu_c=3, the same value as for the single map [2], with the long time behaviour being characterized by extremely sharp distributions, ii) An order parameter can be defined as \psi = \langle x_i(n_{\rm odd/even}) - \overline{x_i} \rangle and is found to vary as \psi \propto (\mu -\mu_c)^{1/2}, with the same exponent as for the single map, and this value is independent of \epsilon for \epsilon \simeq .5, iii) higher periods than four tend to get smeared; the onset of chaotic behaviour depends now on the initial conditions and typically occurs around \mu=3.6. iv) Band merging and rather large stable periodic windows are observed in the chaotic region, the positions of which again depend on the initial conditions (or, equivalently, on i). v) The symbolic (kneading) sequence [2] can be defined with respect to \overline{x}. Then, the approach to chaotic behaviour is signalled by intermittent bursts of out-of-sequence series.

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