Synchronization In Small-world Networks Research Articles

Synchronization of frustrated phase oscillators in the small-world networks

We numerically study the synchronization of an identical population of Kuramoto–Sakaguchi phase oscillators in Watts–Strogatz networks. We find that, unlike random networks, phase shift could enhance the synchronization in small-world networks. We also observe abrupt phase transition with hysteresis at some values of phase shifts in small-world networks, signs of an explosive phase transition. Moreover, we report the emergence of Chimera states at some values of phase shift close to the transition points, which consist of spatially coexisting synchronized and desynchronized domains.

On the possibility of explosive synchronization in small world networks

On the possibility of explosive synchronization in small world networks

Basin stability for burst synchronization in small-world networks of chaotic slow-fast oscillators.

The impact of connectivity and individual dynamics on the basin stability of the burst synchronization regime in small-world networks consisting of chaotic slow-fast oscillators is studied. It is shown that there are rewiring probabilities corresponding to the largest basin stabilities, which uncovers a reason for finding small-world topologies in real neuronal networks. The impact of coupling density and strength as well as the nodal parameters of relaxation or excitability are studied. Dynamic mechanisms are uncovered that most strongly influence basin stability of the burst synchronization regime.

Outer Synchronization of Simple Firefly Discrete Models in Coupled Networks

Synchronization is one of the most important emerging collective behaviors in nature, which results from the interaction in groups of organisms. In this paper, network synchronization of discrete-time dynamical systems is studied. In particular, network synchronization with fireflies oscillators like nodes is achieved by using complex systems theory. Different cases of interest on network synchronization are studied, including for a large number of fireflies oscillators; we consider synchronization in small-world networks and outer synchronization among different coupled networks topologies; for all presented cases, we provide appropriate ranges of values for coupling strength and extensive numerical simulations are included. In addition, for illustrative purposes, we show the effectiveness of network synchronization by means of experimental implementation of coupled nine electronics fireflies in different topologies.

Control of bursting synchronization in networks of Hodgkin-Huxley-type neurons with chemical synapses

Thermally sensitive neurons present bursting activity for certain temperature ranges, characterized by fast repetitive spiking of action potential followed by a short quiescent period. Synchronization of bursting activity is possible in networks of coupled neurons, and it is sometimes an undesirable feature. Control procedures can suppress totally or partially this collective behavior, with potential applications in deep-brain stimulation techniques. We investigate the control of bursting synchronization in small-world networks of Hodgkin-Huxley-type thermally sensitive neurons with chemical synapses through two different strategies. One is the application of an external time-periodic electrical signal and another consists of a time-delayed feedback signal. We consider the effectiveness of both strategies in terms of protocols of applications suitable to be applied by pacemakers.

Noise-induced synchronization in small world networks of phase oscillators

A small-world (SW) network of similar phase oscillators, interacting according to the Kuramoto model, is studied numerically. It is shown that deterministic Kuramoto dynamics on SW networks has various stable stationary states. This can be attributed to the so-called defect patterns in an SW network, which it inherits from deformation of helical patterns in its regular parent. Turning on an uncorrelated random force causes vanishing of the defect patterns, hence increasing the synchronization among oscillators for moderate noise intensities. This phenomenon, called stochastic synchronization, is generally observed in some natural networks such as the brain neural network.

Chaotic phase synchronization in small-world networks of bursting neurons

We investigate the chaotic phase synchronization in a system of coupled bursting neurons in small-world networks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that phase synchronization is largely facilitated by a large fraction of shortcuts, but saturates when it exceeds a critical value. We also study the external chaotic phase synchronization of bursting oscillators in the small-world network by a periodic driving signal applied to a single neuron. It is demonstrated that there exists an optimal small-world topology, resulting in the largest peak value of frequency locking interval in the parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this interval increases with the driving amplitude, but decrease rapidly with the network size. We infer that the externally applied driving parameters outside the frequency locking region can effectively suppress pathologically synchronized rhythms of bursting neurons in the brain.

Synchronization in small-world networks

In this paper we consider complete synchronization in small-world networks of identical Rössler oscillators. By applying a simple but effective dynamical optimization coupling scheme, we realize complete synchronization in networks with undelayed or delayed couplings, as well as ensuring that all oscillators have uniform intensities during the transition to synchronization. Further, we obtain the coupling matrix with much better synchronizability in a certain range of the probability p for adding long-range connections. Direct numerical simulations fully verify the efficiency of our mechanism.

Blinking model and synchronization in small-world networks with a time-varying coupling

The paper proposes a new type of small-world networks of cells with chaotic behavior. This network consists of a regular lattice of cells with constant 2 K-nearest neighbor couplings and time-dependent on–off couplings between any other pair of cells. In each time interval of duration τ such a coupling is switched on with probability p and the corresponding switching random variables are independent for different links and for different times. At each moment, the coupling structure corresponds to a small-world graph, but the shortcuts change from time interval to time interval, which is a good model for many real-world dynamical networks. It is to be distinguished from networks that have randomly chosen shortcuts, fixed in time. Here, we apply the Connection Graph Stability method, developed in the companion paper (“Connection graph stability method for synchronized coupled chaotic systems”, see this issue), to the study of global synchronization in this network with the time-varying coupling structure, in the case where the on–off switching is fast with respect to the characteristic synchronization time of the network. The synchronization thresholds are explicitly linked with the average path length of the coupling graph and with the probability p of shortcut switchings in this blinking model. We prove that for the blinking model, a few random shortcut additions significantly lower the synchronization threshold together with the effective characteristic path length. Short interactions between cells, as in the blinking model, are important in practice. To cite prominent examples, computers networked over the Internet interact by sending packets of information, and neurons in our brain interact by sending short pulses, called spikes. The rare interaction between arbitrary nodes in the network greatly facilitates synchronization without loading the network much. In this respect, we believe that it is more efficient than a structure of fixed random connections.

Connection graph stability method for synchronized coupled chaotic systems

This paper elucidates the relation between network dynamics and graph theory. A new general method to determine global stability of total synchronization in networks with different topologies is proposed. This method combines the Lyapunov function approach with graph theoretical reasoning. In this context, the main step is to establish a bound on the total length of all paths passing through an edge on the network connection graph. In particular, the method is applied to the study of synchronization in rings of 2 K-nearest neighbor coupled oscillators. A rigorous bound is given for the minimum coupling strength sufficient for global synchronization of all oscillators. This bound is explicitly linked with the average path length of the coupling graph. Contrary to the master stability function approach developed by Pecora and Carroll, the connection graph stability method leads to global stability of synchronization, and it permits not only constant, but also time-dependent interaction coefficients. In a companion paper (“Blinking model and synchronization in small-world networks with a time-varying coupling,” see this issue), this method is extended to the blinking model of small-world networks where, in addition to the fixed 2 K-nearest neighbor interactions, all the remaining links are rapidly switched on and off independently of each other.

Phase synchronization in small-world networks of chaotic oscillators

Small-world networks are highly clustered networks with small average distance among the vertices. There are many natural and technological networks that present this kind of connections. We study the phase synchronization of small-world chaotic oscillator networks in this paper. We find that for Rössler oscillator networks, in the synchronous regime, the oscillation phases are locked, while the amplitudes vary chaotically. We further show the dependence of phase synchronization on the network coupling strength, the product of the shortcuts-adding probability and the number of chaotic oscillators, as well as the maximal frequency mismatch.

Chaotic synchronization for dynamical system constructed on star network

Using an example, it was analytically proved that the dynamical systems constructed on a star coupled network is more easy to be synchronized than that in coupled map lattices (CML). The result shows that the synchronization in small-world networks may be realized more easily.