An ω-symmetrically coupled system consisting of identical fractional-order differential systems including chaotic and nonchaotic systems is investigated in this paper. Such a coupled system has, in its synchronous state, a mode decomposition by which the linearized equation can be decomposed into motions transverse to and parallel to the synchronous manifold. Furthermore, the decomposition can induce a sufficient condition on synchronization of the overall system, which guarantees, if satisfied, that a group synchronization is achieved. Two typical numerical examples, fractional Brusselators and the fractional Rossler system, are used to verify the theoretical prediction. The theoretical analysis and numerical results show that the lower the order of the fractional system, the longer the time for achieving synchronization at a fixed coupling strength.

We investigate a general coupled noisy system with time delays, which may be applied to biologically plausible systems for cell-cell communication in a simplified context. The main conclusion is that appropriate noise intensities and coupling strengths are capable of driving the system to be synchronous. We first provide an analytical treatment for the synchronization process, based on the essential phase-locking mechanism, and then derive sufficient conditions which, if satisfied, ensure existence of the synchrony solution. Finally, a multi-cell system with a synthetic gene regulatory network, which contains both intracellular and extracellular noises and time delays, is adopted to demonstrate effects of extracellular noises and couplings on synchronization.

All cell components exhibit intracellular noise due to random births and deaths of individual molecules, and extracellular noise due to environment perturbations. In particular, gene regulation is an inherently noisy process with transcriptional control, alternative splicing, translation, diffusion and chemical modification reactions, which all involve stochastic fluctuations. Such stochastic noises may not only affect the dynamics of the entire system but may also be exploited by living organisms to actively facilitate certain functions, such as cooperative behavior and communication. Results: We have provided a general model and an analytic tool to examine the cooperative behavior of a multi-cell system with both intracellular and extracellular stochastic fluctuations. A multi-cell system with a synthetic gene network is adopted to demonstrate the effects of noises and coupling on collective dynamics. These results establish not only a theoretical foundation but also a quantitative basis for understanding essential roles of noises on cooperative dynamics, such as synchronization and communication among cells.