One-dimensional Coupled Map Lattices Research Articles

Emergence of order in dynamical phases in coupled fractional gauss map

Dynamical behaviour of discrete dynamical systems has been investigated extensively in the past few decades. However, in several applications, long term memory plays an important role in the evolution of dynamical variables. The definition of discrete maps has recently been extended to fractional maps to model such situations. We extend this definition to a spatiotemporal system. We define a coupled map lattice on different topologies, namely, one-dimensional coupled map lattice, globally coupled system and small-world network. The spatiotemporal patterns in the fractional system are more ordered. In particular, synchronization is observed over a large parameter region. For integer order coupled map lattice in one dimension, synchronized periodic states with a period greater than one are not obtained. However, we observe synchronized periodic states with period-3 or period-6 in one dimensional coupled fractional maps even for a large lattice. With nonlocal coupling, the synchronization is reached over a larger parameter regime. In all these cases, the standard deviation decays as power-law in time with the power same as fractional-order. The physical significance of such studies is also discussed.

Li\u2013Yorke chaos on one-dimensional map lattices

In this paper, we consider a special, but important class of one-dimensional coupled map lattices, namely, in which the local dynamics including logistic map as a prototype possesses a snap-back repeller. For smaller coupling strengths, the existence of Li–Yorke scrambled set is proved.

Coupled Map Lattices as Musical Instruments

Abstract We use one-dimensional coupled map lattices (CMLs) to generate sounds that reflect their spatial organization and temporal evolution from a random initial configuration corresponding to uncorrelated noise. In many instances, the process approaches an equilibrium, which generates a sustained tone. The pitch of this tone is proportional to the lattice size, so the CML behaves like an instrument that could be tuned. Among exceptional cases, we provide an example with a metastable strange attractor, which produces an evolving sound reminiscent of drone music.

A NOVEL ALGORITHM FOR IMAGE ENCRYPTION BASED ON COUPLE CHAOTIC SYSTEMS

In this paper, an image encryption algorithm based on couple multiple chaotic systems is presented. It made the one-dimensional Coupled Map Lattice (CML) formed by Skew Tent map as spatiotemporal chaotic system and made its output sequence as the initial value of logistic and meanwhile did iterative of specific times to get the final key sequence, and then did XOR operations with corresponding pixels to finish the encryption. Numerical analysis expresses that this algorithm has large enough space and high security.

Collective behavior of coupled map lattices with different scales of local coupling

In this paper, I present a numerical study on the collective behavior of one-dimensional coupled map lattices with the nearest coupling to different scales for the whole system. Using the maximum Lyapunov exponent as a tool for subsystem and return mapping, I observed several basic patterns of collective behavior and investigated the contrasts between the different scales. To study the mechanism, the system under entirely random perturbations was investigated using the Monte Carlo method and the contrast with the deterministic approach is given. The results show that the response to a random input is complicated and involves the correlation of different signals and taking into consideration the dynamic properties of the system itself.

Control and synchronization of phase-conjugate wave spatiotemporal chaos system driven by CO2 laser

Taking an one-dimensional coupled map lattice as an object, the characteristics of phase-conjugate wave spatiotemporal chaos system are investigated. Based on the Lyapunov stable theorem, the control of phase-conjugate wave spatiotemporal chaos system is achieved, and the multiple phase-conjugate wave spatiotemporal chaos system is driven into the parallel synchronization by the CO2 laser through choosing the coupling parameter. Numerical simulation results show that the coupling parameter has influence on the control and the synchronization speed of the phase-conjugation wave spatiotemporal chaos system. The greater the coupling parameter, the shorter the synchronization timeis.

Scaling properties of invariant densities of coupled Chebyshev maps

We study one-dimensional coupled map lattices consisting of diffusively coupled Chebyshev 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 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 -dependence of the expectation value of any given observable.

Phase transitions in coupled map lattices and in associated probabilistic cellular automata

Analytical tools are applied to investigate piecewise linear coupled map lattices in terms of probabilistic cellular automata. The so-called disorder condition of probabilistic cellular automata is closely related with attracting sets in coupled map lattices. The importance of this condition for the suppression of phase transitions is illustrated by spatially one-dimensional systems. Invariant densities and temporal correlations are calculated explicitly. Ising type phase transitions are found for one-dimensional coupled map lattices acting on repelling sets and for a spatially two-dimensional Miller-Huse-like system with stable long time dynamics. Critical exponents are calculated within a finite size scaling approach. The relevance of detailed balance of the resulting probabilistic cellular automaton for the critical behavior is pointed out.

Controlling localized spatiotemporal chaos using feedback control method

Suppression of localized spatiotemporal chaos observed in one-dimensional coupled map lattice system is achieved using feedback control [P. Parmananda, Yu. Jiang, Phys. Lett. A 231 (1997) 159; P. Parmananda, M. Hildebrand, M. Eiswirth, Phys. Rev. E 56 (1997) 239]. The control is successful both for the frozen random pattern and defect chaotic diffusion pattern. This Letter introduces a new improved feedback control method. To compare with other methods of feedback control, this method can achieve stable state with less iteration steps and more simple calculation process. We prove the stability of the controlled result by calculating maximal Lyapunov exponent. And we also find that this method is robust to small disturbance.

Synchronization universality classes and stability of smooth coupled map lattices

We study two problems related to spatially extended systems: the dynamical stability and the universality classes of the replica synchronization transition. We use a simple model of one-dimensional coupled map lattices and show that chaotic behavior implies that the synchronization transition belongs to the multiplicative noise universality class, while stable chaos implies that the synchronization transition belongs to the directed percolation universality class.

Dynamical properties of the synchronization transition.

We use spreading dynamics to study the synchronization transition (ST) of one-dimensional coupled map lattices (CML's). Recently, Baroni et al. [Phys. Rev. E 63, 036226 (2001)] have shown that the ST belongs to the directed percolation (DP) universality class for discontinuous CML's. This was confirmed by accurate numerical simulations for the Bernoulli map by Ahlers and Pikovsky [Phys. Rev. Lett. 88, 254101 (2002)]. Spreading dynamics confirms such an identification only for random synchronized states. For homogeneous synchronized states the spreading exponents eta and delta are different from the DP exponents but their sum equals the corresponding sum of the DP exponents. Such a relation is typical of models with infinitely many absorbing states. Moreover, we calculate the spreading exponents for the tent map for which the ST belongs to the bounded Kardar-Parisi-Zhang (BKPZ) universality class. The estimation of spreading exponents for random synchronized states is consistent with the hyperscaling relation, while it is inconsistent for the homogeneous ones. Finally, we examine the asymmetric tent map. For small asymmetry the ST remains of the BKPZ type. However, for large asymmetry a different critical behavior appears, with exponents being relatively close to those for DP.

Periodic solutions in one-dimensional coupled map lattices

It is proven that the existence of nonlinear solutions with time period in one-dimensional coupled map lattice with nearest neighbor coupling. This is a class of systems whose behavior can be regarded as infinite array of coupled oscillators. A method for estimating the critical coupling strength below which these solutions with time period persist is given. For some particular nonlinear solutions with time period, exponential decay in space is proved.

Characterization of the stretched-exponential trap-time distributions in one-dimensional coupled map lattices.

Stretched-exponential distributions and relaxation responses are encountered in a wide range of physical systems such as glasses, polymers, and spin glasses. As found recently, this type of behavior occurs also for the distribution function of a certain trap time in a number of coupled dynamical systems. We analyze a one-dimensional mathematical model of coupled chaotic oscillators that reproduces an experimental setup of coupled diode resonators and identify the necessary ingredients for stretched-exponential distributions.

Chaotic strings and standard model parameters

We introduce the so-called chaotic strings (distinguished one-dimensional coupled map lattices underlying the Parisi–Wu approach of stochastic quantization on a small scale) as a possible amendment for ordinary quantization schemes. These strings are strongly self-interacting and exhibit strongest possible chaotic behavior. Constraints on the vacuum energy of the strings fix a certain discrete set of allowed string couplings. We provide extensive numerical evidence that these string couplings numerically coincide with running standard model coupling constants, evaluated at energy scales given by the masses of the known quarks, leptons and gauge bosons. Chaotic strings can thus be used to provide a theoretical argument why certain standard model parameters are realized in nature, others are not, assuming that the a priori free standard model couplings evolve to the minima of the effective potentials. The chaotic string spectrum correctly reproduces the numerical values of the electroweak and strong coupling constants with a precision of 4–5 digits, as well as the (free) masses of the known quarks and leptons with a precision of 3–4 digits. Neutrino mass predictions consistent with present experiments are obtained. The W boson mass also comes out correctly, and a Higgs mass prediction is obtained.

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.

SPATIOTEMPORAL PERIODIC PATTERNS OF A TWO-DIMENSIONAL SYMMETRICALLY COUPLED MAP LATTICES

Aim: The spatiotemporal periodic pattern of a two-dimensional symmetrically coupled map lattice is constructed. Method: Without solving the modeling equations, a series of spatiotemporal periodic orbits in coupled map lattices are deduced by known orbits of one-dimensional coupled map lattices with lower spatial period. The stability of the deduced orbits is analyzed. Results: The L2×L2 Jacobian matrices can be simplified as diagonal matrices of a few 2×2 matrices. Conclusion: The stability of constructed orbits can never be better than that of the original ones.

Relation between coupled map lattices and kinetic ising models

A spatially one-dimensional coupled map lattice possessing the same symmetries as the Miller-Huse model is introduced. Our model is studied analytically by means of a formal perturbation expansion which uses weak coupling and the vicinity to a symmetry breaking bifurcation point. In parameter space four phases with different ergodic behavior are observed. Although the coupling in the map lattice is diffusive, antiferromagnetic ordering is predominant. Via coarse graining the deterministic model is mapped to a master equation which establishes an equivalence between our system and a kinetic Ising model. Such an approach sheds some light on the dependence of the transient behavior on the system size and the nature of the phase transitions.

A coupled map lattice model for dendritic patterns

A coupled map lattice model is proposed as a simple simulation model for growth patterns in a diffusion field. The numerical simulation in a one-dimensional coupled map lattice is consistent with the solution of the Stefan problem. Needle-like and dendritic patterns are obtained in two-dimensional coupled map lattices.

LATTICE THREE-SPECIES MODELS OF THE SPATIAL SPREAD OF RABIES AMONG FOXES

Lattice models describing the spatial spread of rabies among foxes are studied. In these models, the fox population is divided into three-species: susceptible (S), infected or incubating (I), and infectious or rabid (R). They are based on the fact that susceptible and incubating foxes are territorial while rabid foxes have lost their sense of direction and move erratically. Two different models are investigated: a one-dimensional coupled-map lattice model, and a two-dimensional automata network model. Both models take into account the short-range character of the infection process and the diffusive motion of rabid foxes. Numerical simulations show how the spatial distribution of rabies, and the speed of propagation of the epizootic front depend upon the carrying capacity of the environment and diffusion of rabid foxes out of their territory.

Intermediate-range coupling generates low-dimensional attractors deeply in the chaotic region of one-dimensional lattices

Properties of intermediate-range coupling are studied in one-dimensional coupled map lattices (CMLs). Phase diagrams have been constructed which describe the relationship between the range of coupling and coupling strength. A delicate low-dimensional attractor is emerging for non-global interactions in the case of weak coupling, while the leading Lyapunov exponent is a large positive number.