While symmetry has been exploited to analyze synchronization patterns in complex networks, the identification of symmetries in large-size network remains as a challenge. We present in the present work a new method, namely the method of eigenvector-based analysis, to identify symmetries in general complex networks and, incorporating the method of eigenvalue analysis, investigate the formation and transition of synchronization patterns. The efficiency of the proposed method is validated by both artificial and empirical network models consisting of coupled chaotic oscillators. Furthermore, we generalize the method to networks of weighted couplings in which no strict symmetry exists but synchronization clusters are still well organized, with the predications agreeing with the results of direct simulations. Our study provides a new approach for identifying network symmetries, and paves a way to the investigation of synchronization patterns in large-size complex network.
We consider the time required for complete synchronization of two identical one-way coupled vander Pol-Duffing oscillators occurring in the regime of dynamic chaos. The influence of the initial phase differ-ence between oscillators on the duration of the process of complete synchronization has been studied. At a fixedphase of chaotic oscillations of the self-excited drive oscillator, the period of time (past the coupling onset) during which the complete synchronization regime is established depends on the phase of the self-excited responseoscillator
The chaotic synchronization regime in coupled dynamical systems is considered. It has been shown, that the onset of synchronous regime is based on the appearance of the phase relation between interacting chaotic oscillators frequency components of Fourier spectra. The criterion of synchronization of spectral components as well as the measure of synchronization have been discussed. The universal power law has been described. The main results are illustrated by coupled Rössler systems, Van-der-Pol and Van-der-Pol-Duffing oscillators.
We study the constructive role of noises in a Lorenz system with functional delay. The effect of delay can change the dynamics of the system to a chaotic one from its steady state. Induced synchronization with white and colored (red and green) noises are observed between two identical uncoupled systems and enhancement of synchrony is also observed with unidirectional coupling. We investigate both the phenomena in a globally coupled network in the presence of white and color noises.
We investigate the dynamics of an array of logistic maps coupled with random delay times. We report that for adequate coupling strength the array is able to synchronize, in spite of the random delays. Specifically, we find that the synchronized state is a homogeneous steady-state, where the chaotic dynamics of the individual maps is suppressed. This differs drastically from the synchronization with instantaneous and fixed-delay coupling, as in those cases the dynamics is chaotic. Also in contrast with the instantaneous and fixed-delay cases, the synchronization does not dependent on the connec…
Synchronization in a population of oscillators with hyperbolic chaotic phases is studied for two models. One is based on the Kuramoto dynamics of the phase oscillators and on the Bernoulli map applied to these phases. This system possesses an Ott-Antonsen invariant manifold, allowing for a derivation of a map for the evolution of the complex order parameter. Beyond a critical coupling strength, this model demonstrates bistability synchrony-disorder. Another model is based on the coupled autonomous oscillators with hyperbolic chaotic strange attractors of Smale-Williams type. Here a disordered …
We report enhancing of complete synchronization in identical chaotic oscillators when their interaction is mediated by a mismatched oscillator. The identical oscillators now interact indirectly through the intermediate relay oscillator. The induced heterogeneity in the intermediate oscillator plays a constructive role in reducing the critical coupling for a transition to complete synchronization. A common lag synchronization emerges between the mismatched relay oscillator and its neighboring identical oscillators that leads to this enhancing effect. We present examples of one-dimensional open …
We review some recent work on the synchronization of coupled dynamical systems on a variety of networks. When nodes show synchronized behaviour, two interesting phenomena can be observed. First, there are some nodes of the floating type that show intermittent behaviour between getting attached to some clusters and evolving independently. Secondly, two different ways of cluster formation can be identified, namely self-organized clusters which have mostly intra-cluster couplings and driven clusters which have mostly inter-cluster couplings.
The problem of dynamic estimation of all parameters of a model representing chaotic and hyperchaotic systems using information from a scalar measured output is solved. The variational calculus based method is robust in the presence of noise, enables online estimation of the parameters and is also able to rapidly track changes in operating parameters of the experimental system. The method is demonstrated using the Lorenz, Rossler chaos and hyperchaos models. Its possible application in decoding communications using chaos is discussed.
We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of 1-d maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon ana…
We show two examples of noise--induced synchronization. We study a 1-d map and the Lorenz systems, both in the chaotic region. For each system we give numerical evidence that the addition of a (common) random noise, of large enough intensity, to different trajectories which start from different initial conditions, leads eventually to the perfect synchronization of the trajectories. The largest Lyapunov exponent becomes negative due to the presence of the noise terms.
We study the influence of network topology and connectivity on the synchronization properties of chaotic logistic maps, interacting with random delay times. Four different types of topologies are investigated: two regular (a ring-type and a ring-type with a central node) and two random (free-scale Barabasi-Albert and small-world Newman-Watts). The influence of the network connectivity is studied by varying the average number of links per node, while keeping constant the total input that each map receives from its neighbors. For weak coupling the array does not synchronize regardless the topolo…
Generalized chaotic synchronization regime is observed in the unidirectionally coupled one-dimensional Ginzburg-Landau equations. The mechanism resulting in the generalized synchronization regime arising in the coupled spatially extended chaotic systems demonstrating spatiotemporal chaotic oscillations has been described. The cause of the generalized synchronization occurrence is studied with the help of the modified Ginzburg-Landau equation with additional dissipation.
In this paper, we discuss distributed adaptive algorithms for synchronization of complex networks, consensus of multi-agents with or without pinning controller. The dynamics of individual node is governed by generalized QUAD condition. We design new algorithms, which can keep the left eigenvector of the adaptive coupling matrix corresponding to the zero eigenvalue invariant. Based on this invariance, various distributive adaptive synchronization, consensus, anti-synchronization models are given.
We study the dynamics of an ensemble of globally coupled chaotic logistic maps under the action of a learning algorithm aimed at driving the system from incoherent collective evolution to a state of spontaneous full synchronization. Numerical calculations reveal a sharp transition between regimes of unsuccessful and successful learning as the algorithm stiffness grows. In the regime of successful learning, an optimal value of the stiffness is found for which the learning time is minimal.
In this paper, synchronization of fractional order Coullet system with precise and also unknown parameters are studied. The proposed method which is based on the adaptive backstepping, has been developed to synchronize two chaotic systems with the same or partially different attractor. Sufficient conditions for the synchronization are analytically obtained. There after an adaptive control law is derived to make the states of two slightly mismatched chaotic Coullet systems synchronized. The stability analysis is then proved using the Lyapunov stability theorem. It is the privilege of the approa…
We investigate the existence of an optimal interplay between the natural frequencies of a group chaotic oscillators and the topological properties of the network they are embedded in. We identify the conditions for achieving phase synchronization in the most effective way, i.e., with the lowest possible coupling strength. Specifically, we show by means of numerical and experimental results that it is possible to define a synchrony alignment function linking the natural frequencies of a set of non-identical phase-coherent chaotic oscillators with the topology of the Laplacian matrix $L$, the la…
We analyze the critical transitions (a) to endemic states in an SIS epidemiological model, and (b) to full synchronization in an ensemble of coupled chaotic maps, on networks where, at any given time, each node is connected to just one neighbour. In these "monogamous" populations, the lack of connectivity in the instantaneous interaction pattern -that would prevent both the propagation of an infection and the collective entrainment into synchronization- is compensated by occasional random reconnections which recombine interacting couples by exchanging their partners. The transitions to endemic…
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.
Auxiliary system approach and various nearest neighbor methods are widely used to detect generalized synchronization in non-identical coupled systems. These methods generally give contradictory results. Therefore one method alone is not sufficient to predict correct result. We show in this report that it is necessary to apply multiple methods together to come to a conclusion. These methods show a signature of generalized synchronization in diffusively coupled non-identical chaotic parametric excited pendula. But we finally find it to be the almost synchronization. It is achieved when the secon…
When we look at the world around us, we see complex physical systems and emergent phenomena. Emergence occurs when a system is observed to have properties that its parts do not have on their own. These properties or behaviors emerge only when the parts interact in a wider whole. Examples of emergence can vary from the synchronization of pendulum clocks hanging on the same wall to the phenomenon of life as an emergent property of chemistry. One of the most complex systems that exist in nature is the human brain. It contains on average 100 to 200 billion neurons and about 100 trillion synapses c…
Coupling of chaotic oscillators has evidenced conditions where synchronization is possible, therefore a nonlinear system can be driven to a particular state through input from a similar oscillator. Here we expand this concept of control of the state of a nonlinear system by showing that it is possible to induce it to follow a \textit{linear} superposition of signals from multiple equivalent systems, using only partial information from them, through one- or more variable-signal. Moreover, we show that the larger the number of trajectories added to the input signal, the better the convergence of…
We study the dynamical behaviors of this improved memristive neuron model by changing external harmonic current and the magnetic gain parameters. The model shows rich dynamics including periodic and chaotic spiking and bursting, and remarkably, chaotic super-bursting, which has greater information encoding potentials than a standard bursting activity. Based on Krasovskii-Lyapunov stability theory, the sufficient conditions (on the synaptic strengths and magnetic gain parameters) for the chaotic synchronization of the improved model are obtained. Based on Helmholtz's theorem, the Hamilton funct…
We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scale-free and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants generally exhibit poor synchronization. On the other hand, connection delays can actually be conducive to synchronization, so that …
We study numerically the dynamics of a network made of two coupled one-dimensional ensembles of discrete-time systems. The first ensemble is represented by a ring of nonlocally coupled Henon maps, and the second one - by a ring of nonlocally coupled Lozi maps. We find that the network of coupled ensembles can realize all the spatio-temporal structures which are observed both in the Henon map ensemble and in the Lozi map ensemble when uncoupled. Moreover, we reveal a new type of spatiotemporal structure, a solitary state chimera, in the considered network. We also establish and describe the eff…
The effect of the chaotic dynamical states of the agents on the coevolution of cooperation and synchronization in a structured population of the agents remains unexplored. With a view to gaining insights into this problem, we construct a coupled map lattice of the paradigmatic chaotic logistic map by adopting the Watts--Strogatz network algorithm. The map models the agent's chaotic state dynamics. In the model, an agent benefits by synchronizing with its neighbours and in the process of doing so, it pays a cost. The agents update their strategies (cooperation or defection) by using either a st…
