Enhanced synchronizability by structural perturbations

In this Brief Report, we investigate the collective synchronization of a system of coupled oscillators on a Barabási-Albert scale-free network. We propose an approach of structural perturbations aiming at those nodes with maximal betweenness. This method can markedly enhance the network synchronizability, and is easy to realize. The simulation results show that the eigenratio will sharply decrease by one-half when only 0.6% of those hub nodes occur under three-division processes when the network size . In addition, the present study also provides numerical evidence that the maximal betweenness plays a major role in network synchronization.

Synchronization in power-law networks

We consider realistic power-law graphs, for which the power-law holds only for a certain range of degrees. We show that synchronizability of such networks depends on the expected average and expected maximum degree. In particular, we find that networks with realistic power-law graphs are less synchronizable than classical random networks. Finally, we consider hybrid graphs, which consist of two parts: a global graph and a local graph. We show that hybrid networks, for which the number of global edges is proportional to the number of total edges, almost surely synchronize.

Complete synchronization of the noise-perturbed Chua's circuits

In this paper, complete synchronization between unidirectionally coupled Chua's circuits within stochastic perturbation is investigated. Sufficient conditions of complete synchronization between these noise-perturbed circuits are established by means of the so-called LaSalle-type invariance principle for stochastic differential equations. Specific examples and their numerical simulations are also provided to demonstrate the feasibility of these conditions. Furthermore, the results obtained for the coupled Chua's circuits are further generalized to the wide class of coupled systems within stochastic perturbation.

Simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems

In this paper, we define a simple criterion of the synchronization threshold in the set of coupled chaotic systems (flows or maps) with diagonal diffusive coupling. The condition of chaotic synchronization is determined only by two "parameters of order," i.e., the largest Lyapunov exponent and the coupling coefficient. Our approach can be applied for both regular chaotic networks and arrays or lattices of chaotic oscillators with irregular, arbitrarily assumed structure of coupling. The main idea of the synchronization stability criterion is based on linear analysis of the ensembles of simplest dynamical systems. Numerical simulations confirm that such a linear approach approximates the synchronization threshold with high precision.

Stability of a neural network model with small-world connections

Small-world networks are highly clustered networks with small distances among the nodes. There are many biological neural networks that present this kind of connection. There are no special weightings in the connections of most existing small-world network models. However, this kind of simply connected model cannot characterize biological neural networks, in which there are different weights in synaptic connections. In this paper, we present a neural network model with weighted small-world connections and further investigate the stability of this model.

General stability analysis of synchronized dynamics in coupled systems

We consider the stability of synchronized states (including equilibrium point, periodic orbit, or chaotic attractor) in arbitrarily coupled dynamical systems (maps or ordinary differential equations). We develop a general approach, based on the master stability function and Gershgörin disk theory, to yield constraints on the coupling strengths to ensure the stability of synchronized dynamics. Systems with specific coupling schemes are used as examples to illustrate our general method.

Synchronization of chaotic systems with different order

The chaotic synchronization of third-order systems and second-order driven oscillator is studied in this paper. Such a problem is related to synchronization of strictly different chaotic systems. We show that dynamical evolution of second-order driven oscillators can be synchronized with the canonical projection of a third-order chaotic system. In this sense, it is said that synchronization is achieved in reduced order. Duffing equation is chosen as slave system whereas Chua oscillator is defined as master system. The synchronization scheme has nonlinear feedback structure. The reduced-order synchronization is attained in a practical sense, i.e., the difference e=x(3)-x(1)(') is close to zero for all time t> or =t(0)> or =0, where t(0) denotes the time of the control activation.

Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays

We show that the stability surface that governs the synchronization of a large class of arrays of identical oscillators can be probed with a simple array of just three identical oscillators. Experimentally this implies that it may be possible to probe the synchronization conditions of many arrays all at the same time. In the process of developing a theory of the three-oscillator probe, we also show that several regimes of asymptotic coupling can be derived for the array classes, including the case of large imaginary coupling, which apparently has not been explored.

A universal circuit for studying chaotic phenomena

In this paper, an overview of the results on an autonomous chaotic electronic circuit, called Chua’s oscillator, is given. Along with brief descriptions of numerical and analytical investigations on Chua’s oscillator, we present some of its potential applications. The significance of the oscillator for the study of general dynamical systems is discussed.