Pattern evolution in non-synchronizable scale-free networks

Pattern formation and evolution in the desynchronizing process of scale-free complex networks are investigated. Depending on how far the system is away from the synchronizable regime, two types of synchronous patterns are identified, namely, the giant-cluster state (GCS) and the scattered-cluster state (SCS). GCS is observed when a system is immediately outside of the synchronizable regime, where the dynamics undergoes a process of on-off intermittency and the patterns are signatured by the existence of a giant synchronous cluster. As the system leaves away from the synchronizable regime, GCS gradually transforms into SCS, accompanied by the continuous dissolving of the giant cluster. Both the two types of patterns are non-stationary, reflected as the timely changed size and content of the clusters. By introducing a new form of synchronization, the temporal phase synchronization, we investigate the dynamical and statistical properties of these non-stationary patterns. An interesting finding is that the unstable nodes of GCS, i.e. nodes that escape from the giant cluster more frequently, are independent of the coupling strength but are sensitive to the bifurcation types. The intermittent behavior of GCS is analyzed by a theory of snapshot attractors, and the theoretical predications fit the numerical observations qualitatively well.     

Analytic treatment of the network synchronization problem with time delays

Motivated by novel results in the theory of network synchronization, we analyze the effects of nonzero time delays in stochastic synchronization problems with linear couplings in an arbitrary network. We determine analytically the fundamental limit of synchronization efficiency in a noisy environment with uniform time delays. We show that the optimal efficiency of the network is achieved for λτ = π(3/2)/(2√π + 4) ≈ 0.738, where λ is the coupling strength (relaxation coefficient) and τ is the characteristic time delay in the communication between pairs of nodes. Our analysis reveals the underlying mechanism responsible for the trade-off phenomena observed in recent numerical simulations of network synchronization problems.     

Abnormal synchronization in complex clustered networks

Recent research has revealed that complex networks with a smaller average distance and more homogeneous degree distribution are more synchronizable. We find, however, that synchronization in complex, clustered networks tends to obey a different set of rules. In particular, the synchronizability of such a network is determined by the interplay between intercluster and intracluster links. The network is most synchronizable when the numbers of the two types of links are approximately equal. In the presence of a mismatch, increasing the number of intracluster links, while making the network distance smaller, can counterintuitively suppress or even destroy the synchronization. We provide theory and numerical evidence to establish this phenomenon.     

Dual-lag synchronization between coupled chaotic lasers due to path-delay interference

We study experimentally the synchronization dynamics of two semiconductor lasers coupled unidirectionally via two different delayed paths. The emitter laser operates in a chaotic regime characterized by low-frequency fluctuations due to optical feedback and induces a synchronized dynamical activity in the receiver laser, which operates in the continuous-wave regime when uncoupled. Different delays in the two coupling paths lead to the coexistence of two time lags in the synchronized dynamics of the oscillators. This dual-lag synchronization degrades the average synchronization quality of the system of coupled lasers and hinders the transmission of information between them. Numerical simulation results agree with the experimental observations, and allow us to explore this phenomenon in a wide parameter range, and quantify the degree of signal transmission degradation caused by this chaotic path-delay interference.     

Desynchronization by phase slip patterns in networks of pulse-coupled oscillators with delays

Coupling delays may cause drastic changes in the dynamics of oscillatory networks. In the present paper we investigate how coupling delays alter synchronization processes in networks of all-to-all coupled pulse oscillators. We derive an analytic criterion for the stability of synchrony and study the synchronization areas in the space of the delay and coupling strength. Specific attention is paid to the scenario of destabilization on the borders of the synchronization area. We show that in bifurcation points the system possesses homoclinic loops, which give rise to complex long- or quasi-periodic solutions. These newly born solutions are characterized by a synchronous group, from which an oscillator periodically escapes, laps one period, and rejoins. We call such a dynamical regime “phase slip patterns”.     

Coherent cluster atomic ordering in the Fe-27Al intermetallic compound

A method for determining the degree of synchronization of intermittent phase synchronization regime from a time series has been proposed on the basis of estimating the zero conditional Lyapunov exponent. The efficiency of the method has been tested on model systems near the boundary of the appearance of the synchronous regime. The method has been used to determine the degree of synchronization between various regions of the brain of rats of the WAG/Rij line having a genetic predisposition to epilepsy.     

The effect of distributed time-delays on the synchronization of neuronal networks

Here we investigate the synchronization of networks of FitzHugh-Nagumo neurons coupled in scale-free, small-world and random topologies, in the presence of distributed time delays in the coupling of neurons. We explore how the synchronization transition is affected when the time delays in the interactions between pairs of interacting neurons are non-uniform. We find that the presence of distributed time-delays does not change the behavior of the synchronization transition significantly, vis-a-vis networks with constant time-delay, where the value of the constant time-delay is the mean of the distributed delays. We also notice that a normal distribution of delays gives rise to a transition at marginally lower coupling strengths, vis-a-vis uniformly distributed delays. These trends hold across classes of networks and for varying standard deviations of the delay distribution, indicating the generality of these results. So we conclude that distributed delays, which may be typically expected in real-world situations, do not have a notable effect on synchronization. This allows results obtained with constant delays to remain relevant even in the case of randomly distributed delays.     

Master-slave synchronization from the point of view of global dynamics

We present a mathematical framework for the theory of a synchronization phenomenon for dynamical systems discovered by Pecora and Carroll [Phys. Rev. Lett. 64, 821-824 (1990)]. From this perspective, we can synchronize, using a single coordinate, an open dense set of linear systems. We use our insights to synchronize nonlinear systems which were not previously recognized as being synchronizable. (c) 1995 American Institute of Physics.     

Facilitated synchronization of complex networks through a discontinuous coupling strategy

Synchronization stability in complex networks is a topic of theoretical interest and practical importance. Increasing effort has been devoted to the enhancement of synchronizability of networks, or more specifically, the design of synchronizable networks. However, most previous attempts turn the coupling weight/gradient or change the topological interactions, which sometimes is not manageable. In this paper, by adopting a simple kind of discontinuous coupling strategy: the uniform on-off coupling scheme, with on-off period being comparable to the timescale of node dynamics, the problem is solved within the framework of the master stability function. The results show that, this strategy can greatly increase the stable region of synchronization, which means the size of synchronizable networks can be much larger than the traditional case, without any changes of their connections. Furthermore, the synchronization speed can be accelerated considerably, which is even higher than the previous optimal case. The mechanism of the facilitation is revealed and shows that the continuous coupling in fact is one of the worst choices for synchronization in the view of discontinuous coupling strategy. The coupling cost required for synchronization is also examined, which is approximately the same as the continuous coupling.     

Synchronization Transition of Time Delayed Complex Dynamical Networks with Discontinuous Coupling

This paper investigates the synchronization of time delayed complex dynamical networks with periodical on-off coupling. Both the theoretical and numerical results show that, in spite of time delays and on-off coupling, two networks may synchronize if the coupling strength and the on-off rate are large enough. It is shown that, for undirected and strongly connected networks, the upper bound of time delays for synchronization is a decreasing function of the absolute value of the minimum eigenvalue of the adjacency matrix. The theoretical analysis confirms the numerical results and provides a better understanding of the influence of time delays and on-off coupling on the synchronization transition. The influence of random delays on the synchronization is also discussed.     

Delays, connection topology, and synchronization of coupled chaotic maps

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 it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior which the individual units are incapable of producing in isolation.     

Secure communication using a compound signal from generalized synchronizable chaotic systems

By considering generalized synchronizable chaotic systems, the drive-auxiliary system variables are combined suitably using encryption key functions to obtain a compound chaotic signal. An appropriate feedback loop is constructed in the response-auxiliary system to achieve synchronization among the variables of the drive-auxiliary and response-auxiliary systems. We apply this approach to transmit analog and digital information signals in which the quality of the recovered signal is higher and the encoding is more secure.     

多延时互耦合半导体激光器的实时混沌同步特性

针对双延时和三延时互耦合半导体激光器系统,研究了互耦合延时和互耦合强度对实时混沌同步质量的影响,提出了双延时互耦合系统中可将其中一个互耦合延时看作反馈延时的思想,揭示了多延时互耦合半导体激光器系统实时混沌同步条件和规律.研究结果表明,多延时互耦合系统中,某两条双向链路的互耦合延时比值为2,是实现高品质实时混沌同步的基本条件;增大互耦合强度,可以改善实时混沌同步品质,且在较低的等效耦合强度条件下,双延时互耦合系统较三延时互耦合系统更易于实现良好的实时混沌同步.     

Synchronization of diffusionally coupled nonautonomous chaotic pendulums

We consider the phenomenon of mutual synchronization of a chain of diffusionally coupled nonautonomous pendulums in a chaotic regime with increase in the diffusion factor. The system dynamics for different boundary conditions is studied. The boundary of the synchronization existence region is obtained qualitatively. The method of comparison systems and the method of Liapunov's functions are used in this paper.State University, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, Nos. 1–2, pp. 69–73, January–February, 1995.     

Studying the behavior of a nonautonomous Van der Pol oscillator in different time scales with the presence of noise near the synchronization boundary

A new level of organization of the temporal behavior of two coupled complex systems is revealed. We report for the first time the coexistence of two types of intermittent behavior that occurs simultaneously near the boundary of the synchronization regime of coupled chaotic oscillators. This intricate phenomenon was observed both experimentally in a physiological experiment and numerically. The laws for both the distribution and the mean length of the laminar phases versus the control parameter values are analytically deduced. Very good agreement between the theoretical results and the numerically calculated data is shown.     

Some Splitting Theorems for Stably Causal Spacetimes

We prove a splitting theorem for stably causal spacetimes and another splitting theorem for finitely compact spacetimes admitting a proper time synchronizable reference frame.     

Synchronization transitions in coupled time-delay electronic circuits with a threshold nonlinearity

Experimental observations of typical kinds of synchronization transitions are reported in unidirectionally coupled time-delay electronic circuits with a threshold nonlinearity and two time delays, namely feedback delay τ(1) and coupling delay τ(2). We have observed transitions from anticipatory to lag via complete synchronization and their inverse counterparts with excitatory and inhibitory couplings, respectively, as a function of the coupling delay τ(2). The anticipating and lag times depend on the difference between the feedback and the coupling delays. A single stability condition for all the different types of synchronization is found to be valid as the stability condition is independent of both the delays. Further, the existence of different kinds of synchronizations observed experimentally is corroborated by numerical simulations and from the changes in the Lyapunov exponents of the coupled time-delay systems.     

Mutual and intermittent enhancements of synchronization transitions by autaptic and synaptic delay in scale-free neuron networks

In neural networks, there exist both synaptic delays among different neurons and autaptic self-feedback delays in a neuron itself. In this paper, we study synchronization transitions induced by synaptic and autaptic delays in scale-free neuron networks, mainly exploring how these two time delays affect synchronization transitions induced by each other. It is found that the synchronization transitions induced by synaptic (autaptic) delay are intermittently enhanced when autaptic (synaptic) delay is varied. There are optimal autaptic strength and synaptic coupling strength by which the synchronization transitions induced by autaptic and synaptic delays become strongest. The underlying mechanisms are briefly discussed in terms of the relationships of autaptic delay, synaptic delay, and inter-burst interval. These results show that synaptic and autaptic delays could contribute to each other and enhance synchronization transitions in the neuronal networks. This implies that autaptic and synaptic delays could play a vital role for the information transmission in neural systems.     

Projective Synchronization in Time-Delayed Chaotic Systems

For the first time, we report on projective synchronization between two time delay chaotic systems with single time delays. It overcomes some limitations of the previous work, where projective synchronization has been investigated only in finite-dimensional chaotic systems, so we can achieve projective synchronization in infinitedimensional chaotic systems. We give a general method with which we can achieve projective synchronization in time-delayed chaotic systems. The method is illustrated using the famous delay-differential equations related to optical bistability. Numerical simulations fully support the analytical approach.     

Projective Synchronization Between Two Nonidentical Variable Time Delayed Systems

In this paper,we propose a method for the projective synchronization between two different chaotic systems with variable time delays.Using active control approach,the suitable controller is constructed to make the states of two different diverse time delayed systems asymptotically synchronize up to the desired scaling factor.Based on the Lyapunov stability theory,the sufficient condition for the projective synchronization is calculated theoretically.Numerical simulations of the projective synchronization between Mackey-Glass system and Ikeda system with variable time delays are shown to validate the effectiveness of the proposed algorithm.