Coupled Map System Research Articles

Systems of Coupled Piecewise-Linear Maps with Central Element: Stability of a Synchronized State

We investigate the stability of a synchronized state in systems of coupled one-dimensional piecewise-linear maps that, by construction, contain a central element, i.e., an element that interacts with every other (peripheral) element of the system. Two types of systems are considered, namely, systems with and without interaction between peripheral elements. We establish conditions for the strong and weak stability of a synchronized state in systems considered.

Parameter space reduction criteria to search for synchronization domains in coupled discrete systems

Based on analytical considerations, we introduce criteria that enable us to encapsulate the parameter domains for which chaotic synchronization in linearly coupled map systems may be attained. Our aim is to provide means to readily determine parameter regions which preclude synchronization. This results in a significant reduction of parameter space that one needs to explore. Our findings hold for both identical and quasi-identical (small parameter mismatch) maps subjected to unidirectional and bidirectional coupling. As a testing ground we present numerical calculations for the logistic and cubic maps which validate the predictive capability of our approach. Our main contribution relies on the applicability of one of our criteria to experimental situations. Since in real life it is almost impossible to construct two truly identical systems, the results for quasi-identical maps are of particular relevance.

Observational heterarchy enhancing active coupling

Heterarchy and the agent of self-organization are ones of most challengeable topics in complex systems. When the notion of heterarchy is generalized and dilated to general phenomena and is replaced by observational heterarchy, the agent of self-organization (i.e., internal observer carrying measurement) is found even in non-hierarchical system. A usual dynamics in the sense of complex system is articulated into two kinds of dynamics, Intent and Extent dynamics, and the interaction between them, where Intent corresponds to an attribute to a given phenomenon and Extent corresponds to a collection of objects satisfying the phenomenon. We formalize observational heterarchy by introducing pre-equivalence (pre-adjunction) between Intent and Extent, and apply it to a coupled map system. Since the model inherits evoluvability of dynamics, we call it active coupling to distinguish usual (passive) coupling from it. We show that perturbation can modify the dynamics itself and enhance robust behavior opened to emergence in the system of active coupling, and discuss the significance of observational heterarchy in complex systems.

Spectral properties and pattern selection in fractal growth networks

A model for the generation of fractal growth networks in Euclidean spaces of arbitrary dimension is presented. These networks are considered as the spatial support of reaction–diffusion and pattern formation processes. The local dynamics at the nodes of a fractal growth network is given by a nonlinear map, giving raise to a coupled map system. The coupling is described by a matrix whose eigenvectors constitute a basis on which spatial patterns on fractal growth networks can be expressed by linear combination. The spectrum of eigenvalues the coupling matrix exhibits a nonuniform distribution that is reflected in the presence of gaps or niches in the boundaries of stability of the synchronized states on the space of parameters of the system. These gaps allow for the selection of specific spatial patterns by appropriately varying the parameters of the system.

Quasistable states in globally coupled tent map systems.

The characteristics of long lasting but not perpetual chaotic states appear in a wide parameter region in a globally coupled overcritical tent map system are exhibited. The lifetime of the transient state has essential relevance with the system size. In some parameter region, the lifetime saturates at a certain level, while in another region it seems to diverge as the size of the system grows. In order to uncover the dynamical structures in large system size limit, the dynamics of one-body distribution is investigated as an idealized model for the infinitely large coupled map system. Obtained numerical results indicate the correspondence between the characteristics of long transient behavior in finite size system and that of the attractor or the ruin of attractor in the idealized model.

Emergence of patterns in driven and in autonomous spatiotemporal systems

The relationship between a driven extended system and an autonomous spatiotemporal system is investigated in the context of coupled map lattice models. Specifically, a locally coupled map lattice subjected to an external drive is compared to a coupled map system with similar local couplings plus a global interaction. It is shown that, under some conditions, the emergent patterns in both systems are analogous. Based on the knowledge of the dynamical responses of the driven lattice, we present a method that allows the prediction of parameter values for the emergence of ordered spatiotemporal patterns in a class of coupled map systems having local coupling and general forms of global interactions.

Spontaneous structure formation in a network of dynamic elements.

To discover the generic behaviors of dynamic networks, we study a coupled map system with variable coupling strength. It is found that this system spontaneously forms various types of network structure according to the parameter values. Depending on the synchronized or desynchronized motion of unit dynamics, the network structure can be either static or dynamic. The separation of units into two groups, one composed of units with a large number of outgoing connections and the other units with little outgoing connections, is observed in dynamic structure. It is revealed that the mechanism for such separation is a positive feedback between unit and connection dynamics.

Synchronization and intermittency in three-coupled chaotic oscillators.

Synchronization of three-coupled chaotic oscillators was studied with the use of a coupled map system derived for interacting kicked relaxators. Partial synchronization (PS), in which two of the three were synchronized, was observed in addition to complete synchronization. An intermittency associated with the breakdown of the PS, seemingly different from the conventional on-off intermittency, was found. We elucidated the statistics, observing the burst-size distribution, the laminar duration distribution, etc. It was found that the breakdown of the PS generated an anomalous diffusion different from that associated with on-off intermittency.

Robustness and consistency of dynamic clustering in complex systems

The robustness and consistency of dynamic clustering behaviour in standard and adaptive coupled maps was investigated. For illustration purposes, a particular subset of dynamic clustering behaviour was chosen to represent information processing. This behaviour consists of the formation and annihilation of a limited number of synchronization clusters within an episode, which show quasi-periodic activity patterns. Metastable quasi-periodicity and dynamic clustering detection algorithms allow this behaviour to be observed in short segments of a system's orbit. Robustness and consistency of this behaviour were confirmed in compact support regions of the control parameter spaces of standard and extended coupled map systems.

Information transfer and nontrivial collective behavior in chaotic coupled map networks.

The emergence of nontrivial collective behavior in networks of coupled chaotic maps is investigated by means of a nonlinear mutual prediction method. The resulting prediction error is used to measure the amount of information that a local unit possesses about the collective dynamics. Applications to locally and globally coupled map systems are considered. The prediction error exhibits phase transitions at critical values of the coupling for the onset of ordered collective behavior in these networks. This information measure may be used as an order parameter for the characterization of complex behavior in extended chaotic systems.

Dynamics of coupling functions in globally coupled maps: size, periodicity, and stability of clusters.

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 can be characterized. By using this analogy, the dynamics of periodic clusters in systems displaying a constant global coupling are predicted; and for a particular family of coupling functions, it is shown that the stability condition of these clustered states can straightforwardly be derived.

Strange-Nonchaotic-Attractor-Like Behaviors in Coupled Map System**The project supported by National Natural Science Foundation of China, the Nonlinear Science Project of China, and the Foundation of Doctoral Training of Educational Bureau of China

In a coupled map system, an attractor which seems to be strange nonchaotic attractor (SNA) is discovered for nonzero measure in parameter range. The attractor has nonpositive Lyapunov exponent (LE) and discrete structure. We call it strange-nonchaotic-attractor-like (SNA-like) behavior because the size of its discrete structure decreases with the computing precision increasing and the true SNA does not change. The SNA-like behavior in the autonomous system is born when the truncation error of round-off is amplified to the size of the discrete part of the attractor during the long time interval of positive local LE. The SNA-like behavior is easily mistaken for a true SNA judging merely from the largest LE and the phase portrait in double precision computing. In non-autonomous system an SNA-like attractor is also found.

Ising-Type and Other Transitions in One-Dimensional Coupled Map Lattices with Sign Symmetry

We consider a one-dimensional lattice of expanding antisymmetric maps [−1, 1]→[−1, 1] with nearest neighbor diffusive coupling. For such systems it is known that if the coupling parameter e is small there is unique stationary (in time) state, which is chaotic in space-time. A disputed question is whether such systems can exhibit Ising-type phase transitions as e grows beyond some critical value ec. We present results from computer experiments which give definite indication that such a transition takes place: the mean square magnetization appears to diverge as e approaches some critical value, with a critical exponent around 0.9. We also study other properties of the coupled map system.

Riddling, bubbling, and Hopf bifurcation in coupled map systems.

In a simple coupled-map system, we show that the desynchronization of the chaotic attractor occurs through a series of Hopf bifurcations. As a result of the accumulation of these processes, the blowout bifurcation of this coupled map system is a new kind of Hopf bifurcation from a chaotic attractor. Riddling, bubbling, and characters of this particular manifestation of Hopf bifurcation from a chaotic attractor are studied in this paper.

Self-organizing dynamics of coupled map systems

We show that the feedback from the macroscopic dynamics of a system of coupled units can synchronize the dynamics of these units. We studied the dynamics of maps coupled through their variables and control parameters. The feedback adjusted the values of the parameters of each map by using a function that depended on the difference between the Liapunov exponent of each unit and the Liapunov exponent of the mean field of the system. We showed that synchronization of the maps can be achieved under two different conditions: (1) where the maps interact autonomously without a fixed controlling map and (2) where the maps interact nonautomously with a single controlling map with fixed parameters. This method of feedback control may be useful in controlling more general types of parallel distributed systems.

On-Off Intermittency and Riddled Basins of Attraction in a Coupled Map System

The paper examines the appearance of on-off intermittency and riddled basins of attraction in a system of two coupled one-dimensional maps, each displaying type-III intermittency. The bifurcation curves for the transverse destablilization of low periodic orbits embeded in the synchronized chaotic state are obtained. Different types of riddling bifurcation are discussed, and we show how the existence of an absorbing area inside the basin of attraction can account for the distinction between local and global riddling as well as for the distinction between hysteric and non-hysteric blowout. We also discuss the role of the so-called mixed absorbing area that exists immediately after a soft riddling bifurcation. Finally, we study the on-off intermittency that is observed after a non-hysteric blowout bifurcaton.

Structural Criticality of Dynamical Glass State in a Spatially Extended Dynamical System

Many periodic attractors frequently coexist in the phase space of a spatially extended dynamical system, and such a state is called the dynamical glass state [Fujisaka et al., 1993]. We study numerically the dynamical glass state of a coupled map system and find that the measure of the structurally critical attractors (just before the bifurcation points) is large. Hence, the global basin structure of the dynamical glass state is "critical". And the dynamical glass state is observed in a finite region in the parameter space, therefore almost all the trajectories made from random initial states focus on the structurally critical attractors without tuning any control parameters in the region. This interesting new property and the relaxation process from an initial state to one of the final attractors are reported.

Order in the turbulent phase of globally coupled maps

The broad peaks seen in the power spectra of the mean field in a globally coupled map system indicate a subtle coherence between the elements, even in the “turbulent” phase. These peaks are investigated in detail with respect to the number of elements coupled, nonlinearity and global coupling strengths. We find that this roughly periodic behavior also appears in the probability distribution of the mapping, which is therefore not invariant. We also find that these peaks are determined by two distinct components: effective renormalization of the nonlinearity parameter in the local mappings, and the strength of the mean field interaction term. Finally, we demonstrate the influence of background noise on the peaks, which is quite counterintuitive, as they become sharper with increase in strength of the noise, up to a certain critical noise strength.

Algebraic correlations in conserving chaotic systems.

It is argued that for [ital d][gt]1, chaotic systems with a conserved quantity can exhibit generic scale invariance---algebraic decay of spatial correlations without the tuning of external parameters---with exponents analytically calculable from noisy nonchaotic models. Numerical confirmation of these predictions for a specific coupled map system is presented.

Clustered motion in symplectic coupled map systems

Clustered motion of particles is found in Hamiltonian dynamics of symplectic coupled map systems. Particles assemble and move with strong correlation. The motion is chaotic but is distinguishable from random chaotic motion. Lyapunov analysis distinguishes global instability from local fluctuations. Clustered motions have finite lifetime. They have fractal geometric structure in the phase space, as the orbits are trapped to ruins of KAM tori and islands.