The issue of synchronization between dynamical systems has attracted much attention, and the systems with integer-order dynamical networks have been well studied. The synchronous behavior of fractional-order dynamical systems is very interesting and importance, but has rarely been studied. In this paper, we studied the synchronization and anti-synchronization behavior between integer-order dynamical networks and fractional-order dynamical systems via a Takagi-Sugeno fuzzy model. Remarkably, there is synchronous behavior in such a system, and this is dramatically different from the behavior of integer-order dynamical networks. Moreover, we studied the impact of different coupling strengths on the dynamical process of synchronization and robustness of the designed controller to different coupling functions, different dimensions of dynamical equations and different fractional orders. Finally, we propose the theoretical analysis, which coincides well with the numerical simulations of five typical examples.

Synchronization and anti-synchronization of fractional dynamical networks

Stability of small-amplitude periodic solutions near Hopf bifurcations in time-delayed fully-connected PLL networks

On the dynamics of axonal membrane

Dynamics in delayed dierential equations (DDEs) is a well studied problem mainly because DDEs arise in models in many areas of science including biology, physiology, population dynamics and engineering. The change of the nature in the solutions in the parameter space for a network of Phase-Locked Loop oscillators was studied in Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators. Communications in Nonlinear Science and Numerical Simulation, Elsevier BV, 2014, (on-line version), where the existence of Hopf bifurcations for both cases, symmetry-preserving and symmetry-breaking synchronization was well stablished. In this work we continue the analysis exploring the stability of periodic solutions emerging near Hopf bifurcations in the Fixed-point subspace, based on the reduction of the innitedimensional space onto a two-dimensional center manifold. Numerical simulations are presented in order to conrm our analitycal results. Although we explore network dynamics of second-order oscillators, results are extendable to higher order nodes.

Stability of Hopf bifurcations in time-delayed fully-connected PLL networks

A Newman-Watts graph is formed by including random links in a regular lattice. Here, the emergence of synchronization in coupled Newman-Watts graphs is studied. The whole neural network is considered as a toy model of mammalian visual pathways. It is composed by four coupled graphs, in which a coupled pair represents the lateral geniculate nucleus and the visual cortex of a cerebral hemisphere. The hemispheres communicate with each other through a coupling between the graphs representing the visual cortices. This coupling makes the role of the corpus callosum. The state transition of neurons, supposed to be the nodes of the graphs, occurs in discrete time and it follows a set of deterministic rules. From periodic stimuli coming from the retina, the neuronal activity of the whole network is numerically computed. The goal is to find out how the values of the parameters related to the network topology affect the synchronization among the four graphs.

/pdf/on-synchronizing-coupled-retinogeniculocortical-pathways-a-2rc69q9u0x.pdf

On Synchronizing Coupled Retinogeniculocortical Pathways: A Toy Model

Frequency transitions in synchronized neural networks

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Frequency transitions in synchronized neural networks