Synchronization In Oscillator Networks And Its Applications In Biology

Complex networks are ubiquitous,ranging from nature to society,such as the internet,the power grid,genetic regulatory networks,etc.In recent years,many researches have been carried out to discuss the relationship between the topological structure and collective dynamical behaviors of oscillator networks.Therefore,studying network dynamics has been considered as a new and developing subject of Complexity Science.It has attracted much attention from many fields such as physics,mathetics,biology and computer science.In order to explore collective dynamical behaviors in complex networks,researchers built the model of coupled oscillator networks. The researches of collective behaviors in oscillator networks are important not only to explore the mechanisms of those natural phenomena existing widely but also to utilize these mechanisms to serve people's life.This paper combines several new methods to study synchronization of oscillator networks and obtains many original results.These results offer sufficient conditions to relealize synchronization and point out the mechanisms of synchronization.Further investigation is carried out to combine theory with practice by simulating many classical models numerically.The main contents and innovative points of this paper are listed as follows.(1).The study of complete synchronization in coupled oscillator networks can be generalized as the following two aspects.a).As far as the studying objects are concerned,the paper studies coupled oscillator networks with two different types of nonlinear coupling.Most of previous reseaches focused on networks coupled linearly and few researches were carried out to study oscillator networks with nonlinear coupling.Moreover,the study of oscillator networks with nonlinear coupling is interesting and important. For the case that the nonlinear coupling funchtions are increasing functions in the absorbing basin of individual oscillator,this paper gives the sufficient conditions for such type of networks to realize synchronization.For another more general type of coupling flmctions,reseaches are also carried out under the hypothesis of coupling blance,b).As we know,the dynamic behavior of each oscillator is composed of two parts:the inherent dynamic behavior of the uncoupled oscillator and the dynamic behavior influenced by coupling.It happened that two previous methods have difficulties to deal with the stability of these two parts,respectively.The paper combines the two methods by utilizing their merits and solves the respective practical difficulties of the two methods to a certain extent.(2).The paper studies phase synchronization in oscillator networks. Most of previous researches defined an order parameter based on the idea of mean field approach and carried out numerical simulations or experiment survey.Few researches have been carried out theoretically.In this paper, the dynamics of networks is reduced to phase equations by phase reduced method.Analyzing the phase equations through the master stability function method,one can prove theoretically that the oscillators with identical frequency can be in-phase synchronized by weak balanced coupling.(3).Utilizing the theory of Lur'e system,the paper studies the practical synchronization of the system composed by nonidentical cells coupled through quorum sensing.This type of synchronization,which implies the dynamics of each cell is similar but with small difference,is a common kind of phenomenon.The factors influencing the synchronization errors are also pointed out,which can help us understand the mechanisms of collective dynamical behaviors caused by quorum sensing.Numerical simulations are carried out to verify these theoretical results.(4).Frequency synchronization of multicell systems coupled by quorum sensing is also discussed.Two unreasonable hypotheses are often necessary for the previous studies,which aren't adopted here.Without the hypothesis of identical cells,the multicell system is composed by similar but with small difference cells.The second hypothesis,quasi-steady-state approximation, is reasonable in the sense of biology but invalid in the rigorous sense.Exact solution of the signaling molecules in the environment is obtained by the theory of differential equations to replace the hypothesis.At last,a compact summary of this paper is given by combining the advances of the previous researches in this fields.The prospect for future study and the possible difficulties are also given.