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We consider a lattice of weakly coupled expanding circle maps. We construct, via a cluster expansion of the Perron-Frobenius operator, an invariant measure for these infinite dimensional dynamical systems which exhibits space-time-chaos.
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A method to predict the emergence of different kinds of ordered collective behaviors in systems of globally coupled chaotic maps is proposed. The method is based on the analogy between globally coupled maps and a map subjected to an external drive. A vector field which results from this analogy appears to govern the transient evolution of the globally coupled system. General forms of global couplings are considered. Some simple applications are given.
As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension 3, we introduce a $C^2$-open set of diffeomorphisms of whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly $1$. Our proof employs the thicknesses of Cantor sets.
Initially, the logistic map became popular as a simplified model for population growth. In spite of its apparent simplicity, as the population growth-rate is increased the map exhibits a broad range of dynamics, which include bifurcation cascades going from periodic to chaotic solutions. Studying coupled maps allows to identify other qualitative changes in the collective dynamics, such as pattern formations or hysteresis. Particularly, hysteresis is the appearance of different attracting sets, a set when the control parameter is increased and another set when it is decreased -- a multi-stable region. In this work, we present an experimental study on the bifurcations and hysteresis of nearly identical, coupled, logistic maps. Our logistic maps are an electronic system that has a discrete-time evolution with a high signal-to-noise ratio ($sim10^6$), resulting in simple, precise, and reliable experimental manipulations, which include the design of a modifiable diffusive coupling configuration circuit. We find that the characterisations of the isolated and coupled logistic-maps dynamics agrees excellently with the theoretical and numerical predictions (such as the critical bifurcation points and Feigenbaums bifurcation velocity). Here, we report multi-stable regions appearing robustly across configurations, even though our configurations had parameter mismatch (which we measure directly from the components of the circuit and also infer from the resultant dynamics for each map) and were unavoidably affected by electronic noise.
We focus on a linear chain of $N$ first-neighbor-coupled logistic maps at their edge of chaos in the presence of a common noise. This model, characterised by the coupling strength $epsilon$ and the noise width $sigma_{max}$, was recently introduced by Pluchino et al [Phys. Rev. E {bf 87}, 022910 (2013)]. They detected, for the time averaged returns with characteristic return time $tau$, possible connections with $q$-Gaussians, the distributions which optimise, under appropriate constraints, the nonadditive entropy $S_q$, basis of nonextensive statistics mechanics. We have here a closer look on this model, and numerically obtain probability distributions which exhibit a slight asymmetry for some parameter values, in variance with simple $q$-Gaussians. Nevertheless, along many decades, the fitting with $q$-Gaussians turns out to be numerically very satisfactory for wide regions of the parameter values, and we illustrate how the index $q$ evolves with $(N, tau, epsilon, sigma_{max})$. It is nevertheless instructive on how careful one must be in such numerical analysis. The overall work shows that physical and/or biological systems that are correctly mimicked by the Pluchino et al model are thermostatistically related to nonextensive statistical mechanics when time-averaged relevant quantities are studied.
Extended dynamic mode decomposition (EDMD) provides a class of algorithms to identify patterns and effective degrees of freedom in complex dynamical systems. We show that the modes identified by EDMD correspond to those of compact Perron-Frobenius and Koopman operators defined on suitable Hardy-Hilbert spaces when the method is applied to classes of analytic maps. Our findings elucidate the interpretation of the spectra obtained by EDMD for complex dynamical systems. We illustrate our results by numerical simulations for analytic maps.
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