Chun-Hsien Li

SYNCHRONIZATION IN LINEARLY COUPLED DYNAMICAL NETWORKS WITH DISTRIBUTED TIME DELAYS

In this paper, we investigate the global exponential synchronization of linearly coupled dynamical networks with time delays. The time delay considered is of the distributed type and the outer-coupling matrix is not assumed to be symmetric. Employing the Lyapunov functional and matrix inequality techniques, we propose a sufficient condition for the occurrence of global exponential synchronization. Two illustrative examples, the coupled Chuas circuits and the coupled Hindmarsh–Rose neurons, and their numerical simulation results are presented to demonstrate the theoretical analyses.

DIVERSITY OF TRAVELING WAVE SOLUTIONS IN DELAYED CELLULAR NEURAL NETWORKS

This work investigates the diversity of traveling wave solutions for a class of delayed cellular neural networks on the one-dimensional integer lattice ℤ1. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Applying the monotone iteration scheme, we can deduce the existence of monotonic traveling wave solutions provided the templates satisfy the so-called quasi-monotonicity condition. We then consider two special cases of the delayed cellular neural network in which each cell interacts only with either the nearest m left neighbors or the nearest m right neighbors. For the former case, we can directly figure out the analytic solution in an explicit form by the method of step with the help of the characteristic function and then prove that, in addition to the existence of monotonic traveling wave solutions, for certain templates there exist nonmonotonic traveling wave solutions such as camel-like waves with many critical points. For the latter case, employing the comparison arguments repeatedly, we can clarify the deformation of traveling wave solutions with respect to the wave speed. More specifically, we can describe the transition of profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. Some numerical results are also given to demonstrate the theoretical analysis.

A graph approach to synchronization in complex networks of asymmetrically nonlinear coupled dynamical systems

In this paper, we investigate the global exponential synchronization in complex networks of nonlinearly coupled dynamical systems with an asymmetric outer-coupling matrix. Employing the Lyapunov function approach with some graph theory techniques, we improve the so-called connection graph stability method for the synchronization analysis, that was originally developed by Belykh et al. for symmetrically linear coupled dynamical systems, to fit the asymmetrically nonlinear coupled case. We derive some criteria that ensure the nonlinearly coupled as well as linearly coupled dynamical systems to be globally exponentially synchronized. An illustrative example of a regular network with a modular structure of nonlinearly coupled Hindmarsh-Rose neurons is provided. We further consider a small-world dynamical network of nonlinearly coupled Chua’s circuits and demonstrate both theoretically and numerically that the small-world dynamical network is easier to synchronize than the original regular dynamical network. More interestingly, numerical results of a real-world network of the cat cortex modeled by the asymmetrically linear coupled FitzHugh-Nagumo equations are also presented.

EXPONENTIAL SYNCHRONIZATION IN DRIVE-RESPONSE SYSTEMS OF HOPFIELD-TYPE NEURAL NETWORKS WITH TIME DELAYS

In this paper we investigate the drive-response type synchronization of Hopfield-type neural networks with connection time delays for both discrete and distributed cases. By employing the Lyapunov functional method, we propose a sufficient condition to ensure the occurrence of synchronization with exponential rates. This criterion is independent of time delays as well as the types of delay. We prove that the exponential synchronization occurs provided a certain weighted sum of the connection and coupling strengths is negative enough, no matter that the connection time delay is of discrete or distributed case. Numerical examples are provided to illustrate the results.