We show that universality in chaotic elements can be lifted to that in complex systems. We construct a globally coupled Flow lattice (GCFL), an analog of a globally coupled Map lattice (GCML). We find that Duffing GCFL shows the same behavior with GCML; population ratio between synchronizing clusters acts as a bifurcation parameter. Lorenz GCFL exhibits interesting two quasi-clusters in an opposite phase motion. Each of them looks like Will o' the wisp; they dance around in opposite phase.
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.
We study a generalization of globally coupled maps, where the elements are updated with probability $p$. When $p$ is below a threshold $p_c$, the collective motion vanishes and the system is the stationary state in the large size limit. We present the linear stability analysis.
A Duffing oscillator in a certain parameter range shows period-doubling that shares the same Feigenbaum ratio with the logistic map, which is an important issue in the universality in chaos. In this paper a globally coupled lattice of Duffing flows (GCFL), which is a natural extension of the globally coupled logistic map lattice (GCML), is constructed. It is observed that GCFL inherits various intriguing properties of GCML and that universality at the level of elements is thus lifted to that of systems. Phase diagrams of GCFL are determined, which are essentially the same with those of GCML. S…
Bifurcations in a system of coupled maps are investigated. Using symbolic dynamics it is proven that for coupled shift maps the well known space--time--mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state. For coupled non--hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour. The anomalous dependence of fluctuations on the system size is attributed to these bifurcations.
For a system of globally coupled chaotic maps with bistable behaviour we relate the rate function for large deviations in the system size at finite time to dynamical properties of the self consistent Perron-Frobenius operator (SCPFO) that describes the system in the infinite size limit.
We study the synchronization of a linear array of globally coupled identical logistic maps. We consider a time-delayed coupling that takes into account the finite velocity of propagation of the interactions. We find globally synchronized states in which the elements of the array evolve along a periodic orbit of the uncoupled map, while the spatial correlation along the array is such that an individual map sees all other maps in his present, current, state. For values of the nonlinear parameter such that the uncoupled maps are chaotic, time-delayed mutual coupling suppress the chaotic behavior …
We propose a new method to investigate collective behavior in a network of globally coupled chaotic elements generated by a tent map. In the limit of large system size, the dynamics is described with the nonlinear Frobenius-Perron equation. This equation can be transformed into a simple form by making use of the piecewise linear nature of the individual map. Our method is applied successfully to the analyses of stability of collective stationary states and their bifurcations.
We consider a lattice of coupled circle maps, a model arising naturally in descriptions of solid state phenomena such as Josephson junction arrays. We find that the onset of spatiotemporal intermittency (STI) in this system is analogous to directed percolation (DP), with the transition being to an unique absorbing state for low nonlinearities, and to weakly chaotic absorbing states for high nonlinearities. We find that the complete set of static exponents and spreading exponents at all critical points match those of DP very convincingly. Further, hyperscaling relations are fulfilled, leading t…
It is shown how different globally coupled map systems can be analyzed under a common framework by focusing on the dynamics of their respective global coupling functions. We investigate how the functional form of the coupling determines the formation of clusters in a globally coupled map system and the resulting periodicity of the global interaction. The allowed distributions of elements among periodic clusters is also found to depend on the functional form of the coupling. Through the analogy between globally coupled maps and a single driven map, the clustering behavior of the former systems …
There are a few different ways to extend regular nonlinear dynamical systems by introducing power-law memory or considering fractional differential/difference equations instead of integer ones. This extension allows the introduction of families of nonlinear dynamical systems converging to regular systems in the case of an integer power-law memory or an integer order of derivatives/differences. The examples considered in this review include the logistic family of maps (converging in the case of the first order difference to the regular logistic map), the universal family of maps, and the standa…
The phenomenon of turbulence is investigated in the context of globally coupled maps. The local dynamics is given by the Chaté-Manneville minimal map previously used in studies of spatiotemporal intermittency in locally coupled map lattices. New features arise in the globally coupled system; for instance, the transition to turbulence takes place discontinuously at some critical values of the parameters of the system. The critical boundaries between different regimes (laminar, turbulent and fully turbulent) of the system are calculated on the parameter space. Windows of turbulence are present o…
We analyze a system of globally coupled logistic maps with asynchronous updating. We show that its dynamics differs considerably from that of the synchronous case. For growing values of the coupling intensity, an inverse bifurcation cascade replaces the structure of clusters and ordering in the phase diagram. We present numerical simulations and an analytical description based on an effective single-element dynamics affected by internal fluctuations. Both of them show how global coupling is able to suppress the complexity of the single-element evolution. We find that, in contrast to systems wi…
This paper studies the effect of independent additive noise on the synchronous dynamics of large populations of globally coupled maps. Our analysis is complementary to the approach taken by Teramae and Kuramoto [Teramae and Kuramoto, Phys. Rev. E 63:036210, 2001] who pointed out the anomalous scaling properties preceding the loss of coherence. We focus on the macroscopic dynamics that remains deterministic at any noise level and differs from the microscopic one. Using properly defined order parameters, an analytical approach is proposed for describing the collective dynamics in terms of an app…
The affect of demographic stochasticity of a system of globally coupled chaotic maps is considered. A two-step model is studied, where the intra-patch chaotic dynamics is followed by a migration step that coupled all patches; the equilibrium number of agents on each site, $N$, controls the strength of the discreteness-induced fluctuations. For small $N$ (large fluctuations) a period-doubling cascade appears as the coupling (migration) increases. As $N$ grows an extremely slow dynamic emerges, leading to a flow along a one-dimensional family of almost period 2 solutions. This manifold become a …
We revisit the globally coupled map lattice (GCML). We show that in the so called turbulent regime various periodic cluster attractor states are formed even though the coupling between the maps are very small relative to the non-linearity in the element maps. Most outstanding is a maximally symmetric three cluster attractor in period three motion (MSCA) due to the foliation of the period three window of the element logistic maps. An analytic approach is proposed which explains successfully the systematics of various periodicity manifestations in the turbulent regime. The linear stability of th…
We investigate the processes of synchronization and phase ordering in a system of globally coupled maps possessing bistable, chaotic local dynamics. The stability boundaries of the synchronized states are determined on the space of parameters of the system. The collective properties of the system are characterized by means of the persistence probability of equivalent spin variables that define two phases, and by a magnetization-like order parameter that measures the phase-ordering behavior. As a consequence of the global interaction, the persistence probability saturates for all values of the …
We analyze the asymptotic states in the partially ordered phase of a system of globally coupled logistic maps. We confirm that, regardless of initial conditions, these states consist of a few clusters, and they properly belong in the ordered phase of these systems. The transient times necessary to reach the asymptotic states can be very long, especially very near the transition line separating the ordered and the coherent phases. We find that, where two clusters form, the distribution of their sizes corresponds to windows of regular or narrow-band chaotic behavior in the bifurcation diagram of…
Globally coupled maps (GCMs) are prototypical examples of high-dimensional dynamical systems. Interestingly, GCMs formed by an ensemble of weakly coupled identical chaotic units generically exhibit a hyperchaotic 'turbulent' state. A decade ago, Takeuchi et al. [Phys. Rev. Lett. 107, 124101 (2011)] theorized that in turbulent GCMs the largest Lyapunov exponent (LE), $λ(N)$, depends logarithmically on the system size $N$: $λ_\infty-λ(N)\simeq c/\ln N$. We revisit the problem and analyze, by means of analytical and numerical techniques, turbulent GCMs with positive multipliers to show that there…
Systems of a large number N of globally coupled maps have become popular as a relatively simple prototype of high-dimensional dynamics, showing many interesting and typical phenomena like synchronisation, cluster formation and multistability, and having potential applications in systems like Josephson junction arrays or in biophysical models. There exists a wealth of numerical investigations of globally coupled maps. While much progress has been made in the explanation of the macroscopic behaviour of such systems in the limit of infinite N, there is still need for a sound theory about the asym…
In assemblies of globally coupled dynamical units, weak noise perturbing independently the individual units can cause anomalous dispersion in the synchronized cloud of the units in the phase space. When the noise-free dynamics of the synchronized assembly is nonperiodic, various moments of the linear dimension of the cloud as a function of the noise strength exhibit multiscaling properties with parameter-dependent scaling exponents. Some numerical evidence of this peculiar behavior as well as its interpretation in terms of a multiplicative stochastic process with small additive noise is provid…
The unitary evolution maps in closed chaotic quantum graphs are known to have universal spectral correlations, as predicted by random matrix theory. In chaotic graphs with absorption the quantum maps become non-unitary. We show that their spectral statistics exhibit universality at the "soft" edges of the spectrum. The same spectral behavior is observed in many classical non-unitary ensembles of random matrices with rotationally invariant measures.
We study the stability of the fixed-point solution of an array of mutually coupled logistic maps, focusing on the influence of the delay times, $τ_{ij}$, of the interaction between the $i$th and $j$th maps. Two of us recently reported [Phys. Rev. Lett. {\bf 94}, 134102 (2005)] that if $τ_{ij}$ are random enough the array synchronizes in a spatially homogeneous steady state. Here we study this behavior by comparing the dynamics of a map of an array of $N$ delayed-coupled maps with the dynamics of a map with $N$ self-feedback delayed loops. If $N$ is sufficiently large, the dynamics of a map of …
We study the coherent dynamics of globally coupled maps showing macroscopic chaos. With this term we indicate the hydrodynamical-like irregular behaviour of some global observables, with typical times much longer than the times related to the evolution of the single (or microscopic) elements of the system. The usual Lyapunov exponent is not able to capture the essential features of this macroscopic phenomenon. Using the recently introduced notion of finite size Lyapunov exponent, we characterize, in a consistent way, these macroscopic behaviours. Basically, at small values of the perturbation …
We study the synchronization of a coupled map lattice consisting of a one-dimensional chain of logistic maps. We consider global coupling with a time-delay that takes into account the finite velocity of propagation of interactions. We recently showed that clustering occurs for weak coupling, while for strong coupling the array synchronizes into a global state where each element sees all other elements in its current, present state [Physica A {\bf 325} (2003) 186, Phys. Rev. E {\bf 67} (2003) 056219]. In this paper we study the effects of in-homogeneities, both in the individual maps, which are…
Universal scaling laws form one of the central issues in physics. A non-standard scaling law or a breakdown of a standard scaling law, on the other hand, can often lead to the finding of a new universality class in physical systems. Recently, we found that a statistical quantity related to fluctuations follows a non-standard scaling law with respect to system size in a synchronized state of globally coupled non-identical phase oscillators [Nishikawa et al., Chaos $\boldsymbol{22}$, 013133 (2012)]. However, it is still unclear how widely this non-standard scaling law is observed. In the present…
We investigate the emergence of chimera and cluster states possessing asymmetric dynamics in globally coupled systems, where the trajectories of oscillators belonging to different subpopulations exhibit different dynamical properties. In an asymmetric chimera state, the trajectory of an element in the synchronized subset is stationary or periodic, while that of an oscillator in the desynchronized subset is chaotic. In an asymmetric cluster state, the periods of the trajectories of elements belonging to different clusters are different. We consider a network of globally coupled chaotic maps as …
We revisit the globally coupled map lattice (GCML). We use a family of universal 'curves of balance' between two conflicting tendencies in the model- the randomness in each map and the coherence due to an averaging interaction - and we locate the posi-nega switch region in the parameter space of the model. We clarify the mechanism of a basic posi-nega switch in the two-cluster regime, which guarantees no mixing of maps across their mean field in the chaotic transient process. We stress that a special attention must be paid for rounding-off errors; otherwise, there fatally occurs an artifitial …
We propose a systematic approach to the construction of invariant union of polytopes (IUP) in expanding piecewise affine mappings whose linear components are isotropic scalings. The approach relies on using empirical information embedded in trajectories in order to infer, and then to solve, a so-called conditioning problem for some generating collection of polytopes. A conditioning problem consists of a series of requirements on the polytopes' localisation and on the dynamical transitions between these elements. The core element of the approach is a reformulation of the problem as a set of pie…
The Globally Coupled Map Lattice (GCML) is one of the basic model of the intelligence activity. We report that, in its so-called turbulent regime, periodic windows of the element maps foliate and systematically control the dynamics of the model. We have found various cluster attractors. In one type of them, the maps split into several almost equally populated clusters and the clusters mutually oscillate with a period (p) that is the same with the number of clusters (c). We name them as maximally symmetric cluster attractors (MSCA's). The most outstanding are the p3c3 MSCA and its bifurcate. Th…
In this article, we describe a new renormalization-group scheme for analyzing the breakup of invariant tori for Hamiltonian systems with two degrees of freedom. The transformation, which acts on Hamiltonians that are quadratic in the action variables, combines a rescaling of phase space and a partial elimination of irrelevant (non-resonant) frequencies. It is implemented numerically for the case applying to golden invariant tori. We find a nontrivial fixed point and compute the corresponding scaling and critical indices. If one compares flows to maps in the canonical way, our results are consi…
We show that dynamical clustering, where a system segregates into distinguishable subsets of synchronized elements, and chimera states, where differentiated subsets of synchronized and desynchronized elements coexist, can emerge in networks of globally coupled robust-chaos oscillators. We describe the collective behavior of a model of globally coupled robust-chaos maps in terms of statistical quantities, and characterize clusters, chimera states, synchronization, and incoherence on the space of parameters of the system. We employ the analogy between the local dynamics of a system of globally c…
Collective behavior is studied in globally coupled maps with distributed nonlinearity. It is shown that the heterogeneity enhances regularity in the collective dynamics. Low-dimensional quasiperiodic motion is often found for the mean-field, even if each element shows chaotic dynamics. The mechanism of this order is due to the formation of an internal bifurcation structure, and the self-consistent dynamics between the structures and the mean-field. Keywords: Globally Coupled Map with heterogeneity, Collective behavior
The Dynamics of the intermittency maps reveal the existence of resonances phenomena, interesting hybrid states and the orders of the phase transitions in a finite Z(3) spin model in 3D Lattice
We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, $h_\varepsilon$. Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map $\varepsilon\mapsto h_\varepsilon$ is Lipschitz continuous.
The dynamics of globally coupled map lattices can be described in terms of a nonlinear Frobenius--Perron equation in the limit of large system size. This approach allows for an analytical computation of stationary states and their stability. The complete bifurcation behaviour of coupled tent maps near the chaotic band merging point is presented. Furthermore the time independent states of coupled logistic equations are analyzed. The bifurcation diagram of the uncoupled map carries over to the map lattice. The analytical results are supplemented with numerical simulations.
Hamiltonian mixed systems with unbounded phase space are typically characterized by two asymptotic algebraic laws: decay of recurrence time statistics ($γ$) and superdiffusion ($β$). We conjecture the universal exponents $γ=β=3/2$ for trapping of trajectories to regular islands based on our analytical results for a wide class of area-preserving maps. For Hamiltonian mixed systems with bounded phase space the interval $3/2\leqγ_{b}\leq3$ was obtained, given that trapping takes place. A number of simulations and experiments by other authors give additional support to our claims.
We introduce a new method for determining the global stability of synchronization in systems of coupled identical maps. The method is based on the study of invariant measures. Besides the simplest non-trivial example, namely two symmetrically coupled tent maps, we also treat the case of two asymmetrically coupled tent maps as well as a globally coupled network. Our main result is the identification of the precise value of the coupling parameter where the synchronizing and desynchronizing transitions take place.
We consider a general class of intermittent maps designed to be weakly chaotic, i.e., for which the separation of trajectories of nearby initial conditions is weaker than exponential. We show that all its spatio and temporal properties, hitherto regarded independently in the literature, can be represented by a single characteristic function $ϕ$. A universal criterion for the choice of $ϕ$ is obtained within the Feigenbaum's renormalization-group approach. We find a general expression for the dispersion rate $ζ(t)$ of initially nearby trajectories and we show that the instability scenario for w…
Synchronization among globally coupled, chaotic map lattices can be related to stable periodic windows in isolated chaotic maps. This relation provides a simple predictive tool for the understanding of complicated behavior in coupled physical, chemical and biological systems.
A procedure to predict the occurrence of periodic clusters in a system of globally coupled maps displaying a constant mean field is presented. The method employs the analogy between a system of globally coupled maps and a single map driven by a constant force. By obtaining the asymptotic orbits of the driven map, an associated coupling function can be constructed. This function allows to establish a direct connection between both systems. Some applications are shown.
Understanding and predicting uncertain things are the central themes of scientific evolution. Human beings revolve around these fears of uncertainties concerning various aspects like a global pandemic, health, finances, to name but a few. Dealing with this unavoidable part of life is far tougher due to the chaotic nature of these unpredictable activities. In the present article, we consider a global network of identical chaotic maps, which splits into two different clusters, despite the interaction between all nodes are uniform. The stability analysis of the spatially homogeneous chaotic solut…
A mean-field formulation is used to investigate the bifurcation diagram for globally coupled tent maps by means of an analytical approach. It is shown that the period doubling sequence of the single site map induces a continuous family of periodic states in the coupled system. This type of collective motion breaks the ergodicity of the coupled map lattice. The stability analysis suggests that these states are stable for weak coupling strength but opens the possibility for more complicated types of motion in the regime of moderate coupling.
Understanding instabilities in dynamical systems drives to the heart of modern chaos theory, whether forecasting or attempting to control future outcomes. Instabilities in the sense of locally maximal stretching in maps is well understood, and is connected to the concepts of Lyapunov exponents/vectors, Oseledec spaces and the Cauchy--Green tensor. In this paper, we extend the concept to global optimization of stretching, as this forms a skeleton organizing the general instabilities. The `map' is general but incorporates the inevitability of finite-time as in any realistic application: it can b…
We suggest an approach to constructing physical systems with dynamical characteristics of the complex analytic iterative maps. The idea follows from a simple notion that the complex quadratic map by a variable change may be transformed into a set of two identical real one-dimensional quadratic maps with a particular coupling. Hence, dynamical behavior of similar nature may occur in coupled dissipative nonlinear systems, which relate to the Feigenbaum universality class. To substantiate the feasibility of this concept, we consider an electronic system, which exhibits dynamical phenomena intrins…
In the turbulent regime of coupled map lattice with non-local interaction the maps systematically form periodic cluster attractors and their remnants by synchronization due to the foliation of periodic windows of the element map. We examine these periodicity manifestations in three types of coupled map lattices in $D=1, 2, 3$. In the first two, the interaction is all-to-all but the coupling decreases with distance in a power and an exponential law. In the third, the interaction is uniform but cut off sharply. We find that in all three models and in all dimensions periodicity manifests universa…
We revisit the globally coupled map lattice (GCML) and also propose a new extended globally coupled map lattice (EGCML) with an inverse power law interaction. In GCML we clarify the mechanism of the basic posi-nega switch in the two-cluster regime. We show that there is a natural mechanism in GCML which guarantees no mixing of maps across their mean field in the chaotic transient process. In the turbulent regime of GCML there is a prominent period three window. In the correlator analysis we also find a remnant of periodic motion of quasi-clusters with high rate mixing almost everywhere in the …
We investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise-linear Lorenz maps. Riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale, which has serious consequences on the predictability of the final state for such a coupled system. We found that there are wide parameter intervals for which two piecewise-linear Lorenz maps exhibit riddled basins (globally or locally), which indicates that there are riddled basins in coupled Lorenz equations…
Fluctuations of the mean field of a globally coupled dynamical systems are discussed. The origin of hidden coherence is related with the instability of the fixed point solution of the self-consistent Perron-Frobenius equation. Collective dynamics in globally coupled tent maps are re-examined, both with the help of direct simulation and the Perron-Frobenius equation. Collective chaos in a single band state, and bifurcation against initial conditions in a two-band state are clarified with the return maps of the mean-field, Lyapunov spectra, and also the newly introduced Lyapunov exponent for the…
Using a renormalization method, we study the critical behavior for intermittency in two coupled one-dimensional (1D) maps. We find two fixed maps of the renormalization transformation. They all have common relevant eigenvalues associated with scaling of the control parameter of the uncoupled 1D map. However, the relevant ``coupling eigenvalues'' associated with coupling perturbations vary depending on the fixed maps. It is also found that the two fixed maps are associated with the critical behavior in the vicinity of a critical line segment. One fixed map with no relevant coupling eigenvalues …
