Synchronization of two coupled multimode oscillators with time-delayed feedback

Effects of synchronization in a system of two coupled oscillators with time-delayed feedback are investigated. Phase space of a system with time delay is infinite-dimensional. Thus, the picture of synchronization in such systems acquires many new features not inherent to finite-dimensional ones. A picture of oscillation modes in cases of identical and non-identical coupled oscillators is studied in detail. Periodical structure of amplitude death and “broadband synchronization” zones is investigated. Such a behavior occurs due to the resonances between different modes of the infinite-dimensional system with time delay.     

Synchronization phenomena in the system consisting of m coupled Berger plates

The dynamical system arising in the study of nonlinear oscillations of a number of coupled Berger plates is considered. The dependence of the long-time behavior of the trajectories of the system on the properties of the coupling operator is studied. It is shown that the global attractor of the dynamical system is continuous with respect to the coupling parameter γ expressing the intensity of plate interaction. When γ→∞ it converges upper semicontinuously to the attractor of the system generated by the projection of the vector field of the coupled system on the kernel of the coupling operator. For the particular case of 3-diagonal coupling operator the synchronization phenomenon at the level of attractors is stated for large values of γ as well as the absence of synchronization for γ small. The case of cluster synchronization is also considered.     

Synchronization of pendula rotating in different directions

We study the synchronization of a number of pendula mounted on a horizontal beam which can roll on the parallel surface. Under the driving moment, the pendula rotate in different directions: one of them rotates counterclockwise, the rest rotate clockwise. It has been shown that after a transient different types of phase synchronization between pendula can be observed, despite opposite directions of rotations.     

Synchronization criteria for coupled neural networks with interval time-varying delays and leakage delay

This paper considers the synchronization problem for coupled neural networks with interval time-varying delays and leakage delay. By construction of a suitable Lyapunov-Krasovskii’s functional and utilization of Finsler’s lemma, novel delay-dependent criteria for the synchronization of the networks are established in terms of linear matrix inequalities (LMIs) which can be easily solved by various effective optimization algorithms. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Synchronization in time-discrete model of two electrically coupled spike-bursting neurons

Dynamics of the ensemble of two model neurons interacting through electrical synapse is investigated. Both neurons are described by two-dimensional discontinuous map. It is shown that in four-dimensional phase space a chaotic attractor of relaxation type exists corresponding to spike-bursting chaotic oscillations. A new effect of recurrent transitory chaotic oscillations underlies a dynamical mechanism of chaotic bursts formation. It is shown that, under coupling, the transient from chaotic bursts generation into rest state occurs with a time delay. A new characteristic estimating the degree of spike-bursting synchronization is introduced. Dependence of the synchronism degree on the coupling strength is shown for some coupling interval where only activity synchronization occurs. A probabilistic study provides a dynamical explanation of these phenomena.     

Synchronization of an evolving complex hyper-network

An evolving hyper-network model is proposed for better describing some complex systems. A concept of joint degree is introduced, and the evolving mechanism of the hyper-network is given with respect to the joint degree. The hyper-degree distribution of this evolving hyper-network is derived based on a rate equation method and is shown to obey a power law, non-Gaussian distribution. Furthermore, the synchronization in a hyper-network of coupled dynamical systems is investigated for the first time. By calculating the joint degree matrix, several simple yet useful synchronization criteria are obtained and are illustrated numerically in specific examples.     

Chaotic synchronization of two mutually coupled semiconductor lasers for optoelectronic logic gates

A physical model of the fundamental configuration of two mutually coupled semiconductor lasers is presented for logic-gate applications, and the principles of optoelectronic logic computing based on chaotic synchronization or chaotic de-synchronization are defined. Two laser diodes were coupled via injection of each into the opposite laser and became chaotic; our analysis showed that the oscillation derives from chaotic fluctuations after a progression from stability to period-doubling by varying the coupling factor, delay time or detuning. Chaotic synchronization is achieved between the two lasers through the coupling, where we found chaotic and quasi-periodic synchronization regions. Based on the chaotic synchronization system, three optoelectronic logic gates can be implemented by modulating the laser diode current to synchronize or de-synchronize the two chaotic states. Finally, we studied the effects of resynchronization time on logic gate function in a practical implementation of the system. Numerical results show the validity and feasibility of the method.     

Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling

This paper investigates the adaptive synchronization between two nonlinearly delay-coupled complex networks with the bidirectional actions and nonidentical topological structures. Based on LaSalle’s invariance principle, some criteria for the synchronization between two coupled complex networks are achieved via adaptive control. To validate the proposed methods, the unified chaotic system as the nodes of the networks are analyzed in detail, and numerical simulations are given to illustrate the theoretical results.     

新的分数阶混沌系统及其同步

提出一个新的分数阶混沌系统,该系统含有三个参数,三个非线性项.通过理论分析,给出了分数阶混沌系统存在混沌吸引子的必要条件,通过数值仿真给出了混沌吸引子的图像,接着设计自适应同步控制器和参数自适应律,实现分数阶混沌系统的同步,数值仿真的结果表明设计控制器很好的实现了驱动系统和响应系统的同步.     

Synchronization of networks with time-varying couplings

In this paper, we present a review of our recent works on complete synchro-nization analyses of networks of the coupled dynamical systems with time-varying cou-plings. The main approach is composed of algebraic graph theory and dynamic system method. More precisely, the Hajnal diameter of matrix sequence plays a key role in in-vestigating synchronization dynamics and the joint graph across time periods possessing spanning tree is a doorsill for time-varying topologies to reach synchronization. These techniques with proper modification count for diverse models of networks of the cou-pled systems, including discrete-time and continuous-time models, linear and nonlinear models, deterministic and stochastic time-variations. Alternatively, transverse stability analysis of general time-varying dynamic systems can be employed for synchronization study as a special case and proved to be equivalent to Hajnal diameter.     

Synchronization of two coupled systems of J-J type using active sliding mode control

This paper first presents a rigorous mathematical analysis for the active sliding mode control method proposed recently by Haeri and Emadzadeh. Second, the technique is applied to achieve synchronization between two coupled systems of J-J. Numerical simulations are used to verify the above analytical results.     

Synchronization of the self-excited pendula suspended on the vertically displacing beam

We consider the synchronization of a number of self-excited identical pendula hanging on the same beam which can move vertically. We identify different synchronous configurations and investigate their stability. An approximate analytical analysis of the energy balance allows to derive the synchronization conditions, phase difference between pendula and explains the observed types of synchronizations.     

Homoclinic orbits in the dynamics of resonantly driven coupled pendula

We present a plethora of homoclinic and heteroclinic orbits that exist in the phase space of a weakly nonlinear model describing small oscillations of two resonantly driven coupled pendula. These orbits connect equilibria in a resonance band, which is contained in an invariant plane, and is born under perturbation out of a circle of equilibria. Their existence is shown by using a combination of the Melnikov method and singular perturbation techniques.     

Synchronization of time-delay coupled pulse oscillators

We present a detailed study of the dynamics of pulse oscillators with time-delayed coupling. We get the return maps, obtain strict solutions and analyze their stability. For the case of two oscillators, a periodical structure of synchronization regions is found in parameter space, and the regions corresponding to in-phase and antiphase regimes alternate with growth of time delay. Two types of switching between in-phase and antiphase regimes are studied. We also show that for different parameters coupling delay may have synchronizing or desynchronizing effect. Another novel result is that phase locked regimes exist for arbitrary large values. The specificity of system dynamics with large delay is studied.     

Synchronization of coupled chaotic FitzHugh-Nagumo systems

This paper addresses dynamic synchronization of two FitzHugh-Nagumo (FHN) systems coupled with gap junctions. All the states of the coupled chaotic system, treating either as single-input or two-input control system, are synchronized by stabilizing their error dynamics, using simplest and locally robust control laws. The local asymptotic stability, chosen by utilizing the local Lipschitz nonlinear property of the model to address additionally the non-failure of the achieved synchronization, is ensured by formulating the matrix inequalities on the basis of Lyapunov stability theory. In the presence of disturbances, it ensures the local uniform ultimate boundedness. Furthermore, the robustness of the proposed methods is ensured against bounded disturbances besides providing the upper bound on disturbances. To the best of our knowledge, this is the computationally simplest solution for synchronization of coupled FHN modeled systems along with unique advantages of less conservative local asymptotic stability of synchronization errors with robustness. Numerical simulations are carried out to successfully validate the proposed control strategies.     

Statistical convergence of double sequences

The idea of statistical convergence was first introduced by Fast (1951) but the rapid developments were started after the papers of Šalát (1980) and Fridy (1985). Now a days it has become one of the most active area of research in the field of summability. In this paper we define and study statistical analogue of convergence and Cauchy for double sequences. We also establish the relation between statistical convergence and strongly Cesàro summable double sequences.     

Synchronization of singular complex dynamical networks with time-varying delays

This paper considers delay dependent synchronizations of singular complex dynamical networks with time-varying delays. A modified Lyapunov-Krasovskii functional is used to derive a sufficient condition for synchronization in terms of LMIs (linear matrix inequalities) which can be easily solved by various convex optimization algorithms. Numerical examples show the effectiveness of the proposed method.     

Exponential Synchronization of the Linearly Coupled Dynamical Networks with Delays

In this paper,the authors investigate the synchronization of an array of linearly coupled identical dynamical systems with a delayed coupling.Here the coupling matrix can be asymmetric and reducible.Some criteria ensuring delay-independent and delay- dependent global synchronization are derived respectively.It is shown that if the coupling delay is less than a positive threshold,then the coupled network will be synchronized.On the other hand,with the increase of coupling delay,the synchronization stability of the network will be restrained,even eventually de-synchronized.     

Minimal energy configurations of strained multi-layers

We derive an effective plate theory for internally stressed thin elastic layers as are used, e.g., in the fabrication of nano- and microscrolls. The shape of the energy minimizers of the effective energy functional is investigated without a priori assumptions on the geometry. For configurations in two dimensions (corresponding to Euler-Bernoulli theory) we also take into account a non-interpenetration condition for films of small but non-vanishing thickness.