Stability Conditions For Synchronization Research Articles

A novel equivalent reciprocal convex combination technique to sampled‐data master–slave synchronization for chaotic Lur'e systems with time‐varying delays

AbstractFor a class of master–slave (M‐S) systems in chaotic Lur'e systems with time‐varying delays, a sampled‐data synchronization controller is designed, and a new synchronization stability condition is proposed in the form of linear matrix inequality. A novel Lyapunov‐Krasovskii functional (LKF) is constructed by using the looped‐functional method, and the positive definiteness condition of LKF including sampled‐data parts is exchanged with a looping condition by constructing a functional, which should be equal at adjacent sampling times. The M‐S synchronization condition is obtained utilizing the equivalent reciprocal convex combination approach combined with Bessel‐Legendre integral inequality to estimate the LKF derivative. Different from previous methods, due to fully utilizing both nonlinearity and state information at the sampling time via the integral of error systems from the sampling time to current one, the M‐S synchronization condition is less conservative, and obtained sampled‐data controller possesses a longer sampling period. Finally, two numerical examples verify the superiority and validity of the approach.

Parameter Estimation for Hindmarsh–Rose Neurons

In the paper, a new adaptive model of a neuron based on the Hindmarsh–Rose third-order model of a single neuron is proposed. The learning algorithm for adaptive identification of the neuron parameters is proposed and analyzed both theoretically and by computer simulation. The proposed algorithm is based on the Lyapunov functions approach and reduced adaptive observer. It allows one to estimate parameters of the population of the neurons if they are synchronized. The rigorous stability conditions for synchronization and identification are presented.

Stability analysis of intralayer synchronization in time-varying multilayer networks with generic coupling functions.

The stability analysis of synchronization patterns on generalized network structures is of immense importance nowadays. In this article, we scrutinize the stability of intralayer synchronous state in temporal multilayer hypernetworks, where each dynamic units in a layer communicate with others through various independent time-varying connection mechanisms. Here, dynamical units within and between layers may be interconnected through arbitrary generic coupling functions. We show that intralayer synchronous state exists as an invariant solution. Using fast-switching stability criteria, we derive the condition for stable coherent state in terms of associated time-averaged network structure, and in some instances we are able to separate the transverse subspace optimally. Using simultaneous block diagonalization of coupling matrices, we derive the synchronization stability condition without considering time-averaged network structure. Finally, we verify our analytically derived results through a series of numerical simulations on synthetic and real-world neuronal networked systems.

Self-synchronization characteristics of a class of nonlinear vibration system with asymmetrical hysteresis

The self-synchronization characteristics of the two excited motors for the nonlinear vibration system with the asymmetrical hysteresis have been proposed in the exceptional circumstances of cutting off the power supply of one of the two excited motors. From the point of view of the hysteretic characteristics with the asymmetry, a class of nonlinear dynamic model of the self-synchronous vibrating system is presented for the analysis of the hysteretic characteristics of the soil, which is induced by the relation between the stress and the strain in the soil. The periodic solutions for the self-synchronous system with the asymmetrical hysteresis are investigated using nonlinear asymptotic method. The synchronization condition for the self-synchronous vibrating pile system with the asymmetrical hysteresis is theoretical analyzed using the rotor–rotation equations of the two excited motors. The synchronization stability condition also is theoretical analyzed using Jacobi matrix of the phase difference equation of the two excited motors. Using Matlab/Simlink, the synchronous operation of the two excited motors and the synchronous stability operation of the self-synchronous system with the asymmetrical hysteresis are analyzed through the selected parameters. Various synchronous phenomena are obtained through the difference rates of the two excited motors, including the different initial phase and the different initial angular velocity, and so on. Especially, when there is a certain difference in the two excited motors, the synchronous operation of the two excited motors and the synchronous stability operation of the self-synchronous vibrating system with the asymmetrical hysteresis can still be achieved after the power supply of one of the two excited motors has been disconnected. It has been shown that the research results can provide a theoretical basis for the research of the vibration synchronization theory.

Synthesis of oscillators for exact desired periodic solutions having a finite number of harmonics

The synthesis of autonomous oscillators with exact desired periodic steady-state solution is described in this contribution. The vector field of the oscillator differential equation is built up with a conservative and a dissipative part. Both parts are synthesized using an algebraic function describing the desired limit cycle. The desired periodic motion is restricted by a finite numbers of harmonics, whereby the amplitude and the phase shift of every harmonic can be freely chosen, depending on the specific application. Afterwards the synthesis of a periodically driven oscillator with an exact desired periodic response is described. For this purpose, the differential equation of the autonomous oscillator is extended by a state-dependent compensation term that equals the excitation at the steady-state solution. Here the freely definable amplitudes and phase angles of the oscillator motion are restricted by the existence and stability conditions for synchronization.

Synchronous characteristics of a vibration piling system with electromechanical coupling

From the point of view of frequency capture, the nonlinear dynamic models of the self-synchronous vibrating pile system are presented for the analysis of the nonlinear stiffness of the soil, which is induced by the relationship between the nonlinear stress and the strain in the soil. And the nonlinear dynamic models of the self-synchronous vibrating pile system with electromechanical coupling also are presented for the analysis of the pile- soil- electric coupling. The nonlinear characteristics of the vibrating pile in the self-synchronous vibrating pile system with frequency capture are analyzed, and the periodic solutions for the self-synchronous system with frequency capture are investigated using the nonlinear models. The synchronization condition for the self-synchronous vibrating pile system with frequency capture is theoretical analyzed using the rotor-rotation equations of the two-excited motors, and the synchronization stability condition also is theoretical analyzed using Jacobi matrix of the phase difference equation of the two-excited motors. Using Matlab/Simlink, the reverse rotation synchronization of the two-excited motors and the stability of synchronization of the self-synchronous vibrating pile system with electromechanical coupling are analyzed through the selected parameters. The nonlinear phenomena in the self-synchronous vibrating pile system with electromechanical coupling, such as frequency capture and the limit cycles, are reproduced. Various synchronous phenomena are obtained through the difference rates of the two-excited motors, which are induced by the relationship between the phases of the two-excited motors and the rotation speeds of the two-excited motors. It has been shown that the research results can provide theoretical basis for the design and research of the self-synchronous vibratory system.

Synergistic effect of repulsive inhibition in synchronization of excitatory networks.

We show that the addition of pairwise repulsive inhibition to excitatory networks of bursting neurons induces synchrony, in contrast to one's expectations. Through stability analysis, we reveal the mechanism underlying this purely synergistic phenomenon and demonstrate that it originates from the transition between different types of bursting, caused by excitatory-inhibitory synaptic coupling. This effect is generic and observed in different models of bursting neurons and fast synaptic interactions. We also find a universal scaling law for the synchronization stability condition for large networks in terms of the number of excitatory and inhibitory inputs each neuron receives, regardless of the network size and topology. This general law is in sharp contrast with linearly coupled networks with positive (attractive) and negative (repulsive) coupling where the placement and structure of negative connections heavily affect synchronization.

Dynamics, control and information in delay-coupled systems: an overview

Dynamics, control and information in delay-coupled systems: an overview

Design of coupling for synchronization in time-delayed systems

We report a design of delay coupling for targeting desired synchronization in delay dynamical systems. We target synchronization, antisynchronization, lag-and antilag-synchronization, amplitude death (or oscillation death), and generalized synchronization in mismatched oscillators. A scaling of the size of an attractor is made possible in different synchronization regimes. We realize a type of mixed synchronization where synchronization and antisynchronization coexist in different pairs of state variables of the coupled system. We establish the stability condition of synchronization using the Krasovskii-Lyapunov function theory and the Hurwitz matrix criterion. We present numerical examples using the Mackey-Glass system and a delay Rössler system.

Synchronization of Two Self-Synchronous Vibrating Machines on an Isolation Frame

This paper investigates synchronization of two self-synchronous vibrating machines on an isolation rigid frame. Using the modified average method of small parameters, we ded uce the non-dimensional coupling differential equations of the disturbance parameters for the angular velocities of the fo ur unbalanced rotors. Then the stability problem of synchronization for the four unbalanced rotors is converted into the stabili ty problems of two generalized systems. One is the generalized system of the angular velocity disturbance parameters for the four unbalanced rotors, and the other is the generalized system of three phase disturbance parameters. The condition of implementing synchronization is that the torque of frequency capture between each pair of the unbalanced rotors on a vibrating machine is greater than the absolute values of the output electromagnetic torque difference between each pair of motors, and that the torque of frequency capture between the two vibrating machines is greater than the absolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibrating machines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrix is definite positive, and that all the eigenvalues for the generalized s ystem of three phase disturbance parameters have negative real parts. Computer simulations are carried out to verify the results of the theoretical investigation.

Synchronization of Two Self-Synchronous Vibrating Machines on an Isolation Frame

This paper investigates synchronization of two self-synchronous vibrating machines on an isolation rigid frame. Using the modified average method of small parameters, we deduce the non-dimensional coupling differential equations of the disturbance parameters for the angular velocities of the four unbalanced rotors. Then the stability problem of synchronization for the four unbalanced rotors is converted into the stability problems of two generalized systems. One is the generalized system of the angular velocity disturbance parameters for the four unbalanced rotors, and the other is the generalized system of three phase disturbance parameters. The condition of implementing synchronization is that the torque of frequency capture between each pair of the unbalanced rotors on a vibrating machine is greater than the absolute values of the output electromagnetic torque difference between each pair of motors, and that the torque of frequency capture between the two vibrating machines is greater than the absolute value of the output electromagnetic torque difference between the two pairs of motors on the two vibrating machines. The stability condition of synchronization of the two vibrating machines is that the inertia coupling matrix is definite positive, and that all the eigenvalues for the generalized system of three phase disturbance parameters have negative real parts. Computer simulations are carried out to verify the results of the theoretical investigation.

Synchronization of two coupled exciters in a vibrating system of spatial motion

The paper proposes an analytical approach to investigate the synchronization of the two coupled exciters in a vibrating system of spatial motion. Introducing the disturbance parameters for average angular velocity of two exciters, we deduce the non-dimensional coupling equations of angular velocities of two exciters, in which the inertia coupling matrix is symmetric and the stiffness coupling matrix is antisymmetric in a non-resonant vibrating system. The analysis of the coupling dynamic characteristic shows that the coupled cosine effect of the phase angles will cause the torque acting on two motors to limit the increase of phase difference between two exciters as well as sustain its symmetry of two exciters during the running process. It physically explains the peculiarity of self-synchronization of two exciters. The cosine effect of phase angles of the vibrations excited by each exciter will decrease its moment of inertia. The residual moment of inertia of each exciter represents its relative moment of inertia. The stability condition of synchronization of two exciters is that the relative non-dimensional moments of inertia of two exciters are all greater than zero and four times their product is greater than the square of their coefficient of coupled cosine effect of phase angles, which is equivalent to that the inertia coupling matrix is positive definite and all its elements are positive. The numeric results show that the structure of the vibrating system can ensure the stability condition of synchronous operation.

Complete synchronization on multi-layer center dynamical networks

In this paper, complete synchronization of three-layer center networks is studied. By using linear stability analysis approach, several different coupling schemes of three-layer center networks with the Logistic map local dynamics are discussed, and the stability conditions for synchronization are illustrated via some examples.

New eigenvalue based approach to synchronization in asymmetrically coupled networks

Locally and globally exponential stability of synchronization in asymmetrically nonlinear coupled networks and linear coupled networks are investigated in this paper, respectively. Some new synchronization stability criteria based on eigenvalues are derived. In these criteria, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of the sum of the column of the asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability results is that they can be analytically applied to the asymmetrically coupled networks and can overcome the complexity of calculating eigenvalues of the coupling asymmetric matrix. Therefore, these conditions are very convenient to use. Moreover, a necessary condition of globally exponential synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of the coupling matrix.

Dynamical synchronization and stability of complex networks with multi-layer centers

In this paper, the synchronization of some dynamical networks with multi-layer centers are investigated by linear stability analysis approach. A class of two-layer-center networks with logistic map couplings are discussed and the stability conditions for synchronization are given.

New approach to synchronization in asymmetrically coupled networks

Global exponentially synchronization in asymmetrically coupled networks is investigated in this Letter. We extend eigenvalue based method to synchronization in symmetrically coupled network to synchronization in asymmetrically coupled network. A new stability criterion of eigenvalue based is derived. In this criterion, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of sum of column of asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability result is that it can analytical be applied to the asymmetrically coupled networks and overcome the complexity on calculating eigenvalues of coupling asymmetric matrix. Therefore, this condition is very convenient to use. Moreover, a necessary condition of this synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of coupling matrix.

Stability conditions for synchronization of networks with mixed couplings by linear stability analysis

This paper studies some special networks structured with self-organized and driven behavior that coexist in a cluster, moreover, the clusters have dominant intra-cluster and inter-cluster couplings. It is called mixed-system (M-S) here. For this study linear stability analysis was used, and stability conditions for the synchronized state were determined. For the coupling function g(x), the stability state of the network was discussed in two different cases: the linear case with g(x) = x and the nonlinear case with g(x) = f(x). Furthermore, the condition for the emergence of chaos in the networks was given.

Complete synchronization and stability of star-shaped complex networks

Abstract The synchronization of the star-shaped networks are investigated using linear stability analysis. Three different cases of simple-center node or multi-center nodes with Logistic map couplings are discussed and the stability conditions for synchronization are given. The main results are illustrated via a network of eight nodes with four centers.

Synchronized clusters in coupled map networks. II. Stability analysis

We study self-organized and driven synchronization in some simple coupled map networks, namely globally coupled networks and complete bipartite networks, using both linear stability analysis and Lyapunov function approach and determine stability conditions for synchronization. The phase diagrams for the networks studied here have features very similar to the different kinds of structurally similar networks studied in Part I. Lyapunov function approach shows that when any two nodes are in driven synchronization, all the coupling terms in the difference between the variables of these two nodes cancel out, whereas when they are in self-organized synchronization, the direct coupling term between the two nodes adds an extra term while the other couplings cancel out. We also discuss the conditions for the occurrence of a floating node and suggest that the fluctuations of the conditional Lyapunov exponent about zero can be a criterion for its occurrence.