Dynamics Analysis And Synchronization Control Of Two Classes Of Fractional-Order Nonlinear Systems

Fractal dimension is ubiquitous in nature.With the enrichment of fractional calculus's theoretical research results,more and more attention has been paid to the study of fractional-order systems in various fields of natural sciences in recent years.Because fractional-order nonlinear systems describe physical systems more accurately and truly,the related studies and applications are more extensive.In this thesis,the stability,dynamical behaviors and synchronization control of two classes of fractional order nonlinear systems are studied.Some theoretical results have been obtained.That is to say,the stability criteria of each system are obtained,the bifrcation conditions of each system are given,the dynamic characteristics of the system are analyzed when the bifurcation occurs,and the synchronous control problem of three-dimensional fractional-order dynamic system is discussed.The main contents of this thesis are as follows:In the first chapter,this thesis briefly describes the research background of the subject,and summarizes the research status at home and abroad as well as the significance of the research.The basic knowledge of fractional calculus used in this thesis is introduced.In the second chapter,the dynamical characteristics and synchronization control of a class of generalized fractional Langford system are studied.Firstly,based on the definition and properties of Caputo-type fractional differential,the model of integer-order Langford system is extended to fractional order.According to Lyapunov stability theorem,the stability criteria of fractional order Langford system at zero equilibrium point and non-zero equilibrium point are given.Then the Hopf bifurcation behaviors of the system are studied with the order as the critical value,and the dependence of the order on the bifurcation behaviors at equilibrium points are analyzed.Secondly,the synchronization problem of fractional-order Langford system is studied by using drive-response synchronization method,linear control method and Lyapunov function method.The synchronization control between fractional order drive-response is realized and the conditions of synchronization are given.Finally,the correctness of the results is verified by numerical examples and simulation.In the third chapter,the stability and Hopf bifurcation of a class of fractional complex-valued neural networks with time delays are analyzed.Firstly,based on the definition and properties of Caputo fractional derivative,under certain assumptions,the time-delayed fractional complex-valued neural networks are transformed into the time-delayed fractional real-valued neural network for analysis,and according to linearization method,Laplace transform and stability theory,the stability criteria of the system are obtained.Then the stability and Hopf bifurcation behavior of two-dimensional time-delayed fractional-order complex-valued neural network system are further obtained.The delay is taken as the critical value of bifurcation occurrence,and the influence of delay on bifurcation is obtained.The relationship between fractional order and bifurcation point is discussed by numerical example.Finally,the validity of the conclusions is verified by numerical simulation results.In the fourth chapter,the main contents of this thesis are summarized,and the future work is discussed are prospected,so as to further study the dynamical characteristics of fractional order systems and their practical applications.