Narrow Results

In this paper, we discuss the stability and pattern formation issues of a spatiotemporal discrete system based on the modified Klausmeier model. We begin by constructing the corresponding coupled map lattices model. Then the existence and stability analysis is employed to derive the prerequisites for a stable homogeneous stationary state. Through the center manifold theorem and bifurcation theory, the threshold parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are individually determined. Based on the analysis of bifurcation, four pattern formation mechanisms are presented. Finally, we simulate the corresponding results numerically. The simulations exhibit rich dynamical behaviors, such as period-doubling cascades, invariant cycles, periodic windows, chaos, and rich Turing patterns. Four pattern formation mechanisms give rise to rich and complex patterns, including mosaics, spots, circles, spirals and cyclic fragmentation. The analysis and findings from this study enhance our comprehension of the intricate relationships among bifurcation, chaos and pattern formation for the spatiotemporal discrete Klausmeier model.

Many realistic networked systems may face sustained attack over a sequence of time. In this paper, we propose a cascading failure model in coupled map lattices (CMLs) with sustained attack, where a number of crucial nodes will be attacked one by one at each time step. The effects of four attacking strategies: high-degree strategy (HDS), high-clustering coefficient strategy, high-closeness strategy and high-betweenness strategy are compared. This shows that the performance of HDS is better than that of other attacking strategies when the value of the outside attack is small. The effectiveness of HDS on Watts–Strogatz (WS) small-world networks and two real-world networks is extensively investigated. The results indicate that increasing the value of the rewiring probability of WS networks can make the network more robust to resist sustained attack. The sparser the network structure is, the faster the diffusion velocity of cascading failures is. A smaller coupling strength in CMLs can restrain the diffusion velocity of cascades. We also extend our CML-based model with sustained attack from one node strategy (ONS) to two nodes strategy (TNS) and compare the effect of ONS and TNS with different values of the outside attack. It is found that ONS outperforms TNS for small values of the outside attack, but TNS would be a better choice when the outside attack and the time step are all large. Our work will provide new insights into the problem of protecting complex networks against cascading failures with respect to malicious attack.

It was observed in earlier studies, that the mean field of globally coupled maps evolving under synchronous updating rules violated the law of large numbers, and this remarkable result generated widespread research interest. In this work we demonstrate that incorporating increasing degrees of asynchronicity in the updating rules rapidly restores the statistical behavior of the mean field. This is clear from the decay of the mean square deviation of the mean field with respect to lattice size N, for varying degrees of asynchronicity, which shows 1/N behavior upto very large N even when the updating is far from fully asynchronous. This is also evidenced through increasing 1/f² behavior regimes in the power spectrum of the mean field under increasing asynchronicity.

In this paper we present a numerical study of invariant tori in a lattice of coupled logistic maps. In particular, we are interested in bifurcations leading to chaos. Here we consider six different examples of tori breakdown: two of them completely confirm the theory of Afraimovich and Shilnikov, while the others appear peculiar to the model.

We observe purely spatial intermittency accompanied by temporally periodic behavior in an inhomogeneous lattice of coupled logistic maps where the inhomogenity appears in the form of different values of the map parameters at distinct sites. Linear analysis shows that the spatial intermittency appears in the neighborhood of tangent period doubling bifurcation points. The intermittency near the bifurcation points is associated with a power-law distribution for the laminar lengths. The scaling exponent ζ for the laminar length distribution is obtained.

The behavior of two-dimensional coupled map lattices is studied with respect to the global stabilization of unstable local fixed points without external control. It is numerically shown under which circumstances such inherent global stabilization can be achieved for both synchronous and asynchronous updating. Two necessary conditions for inherent global stabilization are derived analytically.

A comparison between polynomial and wavelet expansions for the identification of coupled map lattice (CML) models for deterministic spatiotemporal dynamical systems is presented in this paper. The pattern dynamics generated by smooth and nonsmooth nonlinear maps in a well-known two-dimensional CML structure are analyzed. By using an orthogonal feedforward regression algorithm (OFR), polynomial and wavelet models are identified for the CML's in chaotic regimes. The quantitative dynamical invariants such as the largest Lyapunov exponents and correlation dimensions are estimated and used to evaluate the performance of the identified models.

Individual sites in spatially extended systems of coupled identical maps may exhibit chaotic behavior even if their intrinsic (local) dynamics is regular and stable. For this to happen it is imperative that the spatial interactions are sufficiently strong. So far, this scenario of generating chaos from simple local dynamics has been established rigorously only for special, very narrow classes of local maps. The present article largely extends previous results by showing that the corresponding mechanism of peak-crossing is in fact very general and robust: whenever the local map is sufficiently expanding and exhibits a horseshoe then the emergence of spatial intermittency will be observed in the form of chaotically oscillating sites surrounded by quasi-regular clusters. The results firmly establish peak-crossing as a natural scenario on the route to spacetime chaos.

We generalize the exact solution to the Bernoulli shift map. Under certain conditions, the generalized functions can produce unpredictable dynamics. We use the properties of the generalized functions to show that certain dynamical systems can generate random dynamics. For instance, the chaotic Chua's circuit coupled to a circuit with a noninvertible I–V characteristic can generate unpredictable dynamics. In general, a nonperiodic time-series with truncated exponential behavior can be converted into unpredictable dynamics using noninvertible transformations. Using a new theoretical framework for chaos and randomness, we investigate some classes of coupled map lattices. We show that, in some cases, these systems can produce completely unpredictable dynamics. In a similar fashion, we explain why some well-known spatiotemporal systems have been found to produce very complex dynamics in numerical simulations. We discuss real physical systems that can generate random dynamics.

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

A sufficient condition for global synchronization in coupled map lattices (CML) with translation invariant coupling and arbitrary individual map is proved. As in [Jost & Joy, 2001] where CML with reflection invariant couplings are considered, the condition only involves the linearized dynamics in the diagonal, namely for all points in the diagonal, the derivative must be contractive in all transverse directions. In addition to this result, a (weaker) condition that ensures the CML attractor to be composed of either 2-periodic or constant configurations, is also obtained.

Particle swarm optimization (PSO) is introduced to implement a new constructive learning algorithm for training generalized cellular neural networks (GCNNs) for the identification of spatio-temporal evolutionary (STE) systems. The basic idea of the new PSO-based learning algorithm is to successively approximate the desired signal by progressively pursuing relevant orthogonal projections. This new algorithm will thus be referred to as the orthogonal projection pursuit (OPP) algorithm, which is in mechanism similar to the conventional projection pursuit approach. A novel two-stage hybrid training scheme is proposed for constructing a parsimonious GCNN model. In the first stage, the orthogonal projection pursuit algorithm is applied to adaptively and successively augment the network, where adjustable parameters of the associated units are optimized using a particle swarm optimizer. The resultant network model produced at the first stage may be redundant. In the second stage, a forward orthogonal regression (FOR) algorithm, aided by mutual information estimation, is applied to refine and improve the initially trained network. The effectiveness and performance of the proposed method is validated by applying the new modeling framework to a spatio-temporal evolutionary system identification problem.

We study the effects of parametric noise on a lattice network, which is locally modeled by a two-dimensional Rulkov map. We conclude that at some intermediate noise intensity, parametric noise can induce ordered circular patterns, which indicates the appearance of spatiotemporal coherence resonance in the studied lattice. With the observation of coherence-like manner in linear spatial cross-correlation, the coherence phenomena can be analyzed quantitatively.

In this paper, a new method for controlling spatiotemporal chaos in discrete-time spatially extended systems modeled by coupled map lattices is proposed. This method is based on quasi-sliding mode using a Lyapunov function. This method enables the system to drive the chaotic motion toward any desired trajectory via a sliding surface in the error space. The controller also guarantees finite time convergence of the state trajectory. The main advantage of this method is its robustness with respect to additive uncertainties and its applicability for all types of coupled map lattices. A diffusively coupled map lattice is used as an example to demonstrate the method. Simulation results reveal the robustness and the effectiveness of the method in controlling spatiotemporal chaos in coupled map lattices.

We show that the synchronized states of two systems of identical chaotic maps subject to either, a common drive that acts with a probability p in time or to the same drive acting on a fraction p of the maps, are similar. The synchronization behavior of both systems can be inferred by considering the dynamics of a single chaotic map driven with a probability p. The synchronized states for these systems are characterized on their common space of parameters. Our results show that the presence of a common external drive for all times is not essential for reaching synchronization in a system of chaotic oscillators, nor is the simultaneous sharing of the drive by all the elements in the system. Rather, a crucial condition for achieving synchronization is the sharing of some minimal, average information by the elements in the system over long times.

This paper proposes decentralized delayed feedback control for coupled map lattices on irregular network topologies. Linear stability analysis provides a systematic procedure for designing controllers that does not depend on the number of sites or network topology. To reduce the long waiting time for stabilization, we demonstrate two techniques by numerical simulations: delayed feedback destabilization and time-varying coupling-strength. Furthermore, the robustness of the controlled state against external disturbances is investigated.

In this paper, we consider chaotic synchronization in coupled map lattices (CMLs) with periodic boundary conditions. We give a rigorous proof of the occurrence of synchronization for 1D such CMLs with lattice size n = 5 for suitable parameters in the chaotic regime by Lyapunov method.

In previous papers [Isagi et al., 1997; Satake & Iwasa, 2000], a forest model was proposed. The authors demonstrated numerically that the mature forest could possibly exhibit annual reproduction (fixed point synchronization), periodic and chaotic synchronization as the energy depletion constant $d$ is gradually increased. To understand such rich synchronization phenomena, we are led to study global dynamics of a piecewise smooth map $f_{d,\beta }$ containing two parameters $d$ and $\beta$. Here $d$ is the energy depletion quantity and $\beta$ is the coupling strength. In particular, we obtain the following results. First, we prove that $f_{d,0}$ has a chaotic dynamic in the sense of Devaney on an invariant set whenever $d>1$, which improves a result of [Chang & Chen, 2011]. Second, we prove, via the Schwarzian derivative and a generalized result of [Singer, 1978], that $f_{d,\beta }$ exhibits the period adding bifurcation. Specifically, we show that for any $\beta >0$, $f_{d,\beta }$ has a unique global attracting fixed point whenever $d\le \frac{1}{(\beta +1){(\frac{\beta +1}{\beta +2})}^{\beta }}$ ($<1$) and that for any $\beta >0$, $f_{d,\beta }$ has a unique attracting period $k+1$ point whenever $d$ is less than and near any positive integer $k$. Furthermore, the corresponding period $k+1$ point instantly becomes unstable as $d$ moves pass the integer $k$. Finally, we demonstrate numerically that there are chaotic dynamics whenever $d$ is in between and away from two consecutive positive integers. We also observe the route to chaos as $d$ increases from one positive integer to the next through finite period doubling.

We investigate local interaction between logistic maps in an inhomogeneous lattice numerically with respect to an inhomogeneity parameter $\gamma$ and the coupling constant $ϵ$. In our model, the inhomogeneity appears in the form of different values of the map parameter at different sites. The phase diagram of the model in the $\gamma$–$ϵ$ plane gives seven qualitatively different patterns. These are: synchronized patterns, steady (fixed in time) patterns with spatial period-two, spatially chaotic together with temporally periodic patterns, spatially coherent accompanied with temporally quasi-periodic patterns, spatial intermittency with temporal period-two patterns, spatiotemporal intermittency patterns or spatiotemporal chaotic patterns. Our system exhibits, tangent bifurcations in the transition from synchronized patterns to steady patterns, period-doubling bifurcation from steady patterns to temporal periodic patterns and Neimark–Sacker (Hopf) bifurcation from steady patterns to temporal quasi-periodic patterns. The system also shows the possibility of multistable attractors and the phenomena of hysteresis for some parameter values. We identify our results using techniques such as time series, space-time plot, Fourier transform, bifurcation diagram, stability analysis.

This paper proposes a novel Switching Chaotic Coupled Map Lattices (SCCML) system based on Elementary Cellular Automata (ECA) to address issues found in Coupled Map Lattices (CML) systems such as period windows, poor lattice correlation, and limited chaotic behavior parameters. The system uses a bidirectional coupling method and introduces two ECA with different rules to continuously alter the indexes of the coupled lattices and the underlying chaotic mapping. This approach effectively mitigates correlation issues in CML systems. At the same time, the iterative results of ECA are cascaded with the chaotic mapping as a perturbation of the SCCML system to extend the parameter range of chaotic behavior. The Kolmogorov–Sinai entropy density and universality, bifurcation diagrams, uniformity and correlation are analyzed to investigate the dynamical properties of the proposed system. Theoretical analysis and numerical simulations demonstrate that, compared with other novel spatiotemporal chaotic systems, the proposed system eliminates periodic windows, expands the chaotic behavior parameter range, and decreases the correlation between different lattices. The above experimental results demonstrate the proposed system’s excellent potential for cryptosystems and secure communication. Furthermore, an Encrypted Code Index Modulation (E-CIM) scheme based on reversible ECA encryption is proposed to address the problems of limited pseudo-random sequence resources and low spectrum utilization in direct sequence spread spectrum systems. The proposed scheme uses a chaotic sequence with good correlation and randomness generated by the SCCML spatiotemporal chaos system as the spreading code sequence. In the scheme, the information bits are divided into two parts, namely modulation bits and mapping bits. The modulation bits are mapped to points on the modulation constellation diagram. The mapping bits are encrypted using reversible ECA, and the mapped bits before and after encryption are mapped as spreading sequences with different indexes. Simulation and analysis show that under the same spectrum efficiency conditions, the BER performance of E-CIM is superior to that of the Code Index Modulation (CIM) and Generalized Code Index Modulation (GCIM) schemes by about 2–4dB and superior to that of the Nonorthogonal Code Shift Keying and Code Index Modulation (N-CSK-CIM) scheme by about 0.5dB in an additive Gaussian white noise channel when the BER is 10${}^{-5}$.