Synchronization In Time-delayed Chaotic Systems And Dynamical Behaviors On Complex Networks

In this thesis, we study the synchronization in time-delayed chaotic systems, synchronization on complex dynamical networks, and opinion dynamicson complex networks. The research focuses mainly on five parts:First, we report on projective synchronization in time-delayed chaotic systems. We observe projective synchronization in a delay-differential system related to the optical bistable or hybrid optical device, which extends the scopeof projective synchronization from finite-dimensional to infinite-dimensionalchaotic systems. Based on the Lyapunov stability theory, we give a generalmethod with which we can achieve projective synchronization in time-delayedchaotic systems. The method is illustrated using the famous delay-differentialequations related to the Logistic map. Numerical simulations are given todemonstrate the effectiveness of this method. This method can be widely usedto a glass of time-delayed chaotic systems related to optical bistable or hybridoptical device, e.g., Vall(?)e model, Ikeda model, sine-square model, Mackey-Glass model, etc.Second, generalized projective synchronization in time-delayed chaoticsystems is studied. We observe generalized projective synchronization betweentwo identical time delay chaotic systems with single time delay. It extends therealization of generalized projective synchronization from finite-dimensional toinfinite-dimensional chaotic systems. Based on the Lyapunov stability theory,we give a theoretical analysis using time-delayed chaotic systems related to optical bistable or hybrid optical device and derive the realization condition. Thecorrectness of theoretical analysis is demonstrated by using Ikeda model and thetime-delayed chaotic systems related to Logistic map. The method can be applied to other time-delayed chaotic systems related to opitical bistable device.The method allows us to arbitrarily direct the scaling factor onto any desired value.Third, the collective synchronization of a system of coupled logistic mapson random community networks is investigated. It is found that the synchronizability of the community network is affected by two factors when the size of thenetwork and the number of connections are fixed. One is the number of communities denoted by the parameter m, and the other is the ratioσdefined as theconnection probability p of each pair of nodes within each community to theconnection probability q of each pair of nodes among different communities.Theoretical analysis and numerical results indicate that larger m and smallerσare the key to enhance network synchronizability. We also testify synchronousproperties of the system by analyzing the largest Lyapunov exponents of thesystem.Fourth, we study projective- anticipating synchronization, projective synchronization, and projective-lag synchronization of time-delayed chaotic systems on random networks. We relax some limitations of previous work, whereprojective-anticipating synchronization and projective-lag synchronization canbe achieved only on two coupled chaotic systems. In this paper, we realizeprojective-anticipating synchronization and projective-lag synchronization oncomplex dynamical networks composed of a large number of interconnectedcomponents. At the same time, although previous work studied projective synchronization on complex dynamical networks, the dynamics of single node isLorenz system which has partially linear characteristics in their work. In ourwork, the dynamics of the nodes of the complex networks are time-delayedchaotic systems without the limitation of the partial linearity. So it can be considered as an extension of the dynamics of each individual node from partiallylinear chaotic systems to nonpartially linear chaotic systems, or an extensionfrom finite-dimensional to infinite-dimensional chaotic systems. Based on theLyapunov stability theory and Gerschg(o|¨)rin disk theory, we suggest a generic method to achieve the projective-anticipating synchronization, projective synchronization, and projective-lag synchronization of time-delayed chaotic systems on random dynamical networks, and we find both its sufficent stabilityand existence conditions. The validity of the proposed method is demonstratedand verified by examining specific examples using Ikeda and Mackey-Glasssystems on Erd(o|¨)s-R(?)nyi networks.Finally, we investigate the influence of average degree of network onthe threshold q_c of an order-disorder transition in opinion dynamics. We consider a variant of majority rule (VMR) model on WS network and BA networkwith varying average degree. Using Monte Carlo simulations and finite-sizescaling analysis, we find the relation between and q_c The threshold q_c ofVMR model is greatly affected by the .When is small, q_c increases obviously with the increase of ;A larger value of leads to a larger valueq_c which similar to mean-field solution. We obtain critical exponentsβ/ v,γ/vand l/v for several values of average degree .We also give characteristicsof these exponents and make a comparation with other opinion models.