Dynamics Analysis And Synchronization Of Fractional-order Multi-stable Chaotic Systems

Calculus is an important branch of mathematics that provides a framework for modeling systems that are undergoing change and provides a way to infer mathematical model predictions.Fractional calculus is an ancient mathematical concept that has a history of more than 300 years.Compared with integer-order calculus,fractional-order calculus can much more accurately describe the essential phenomena of the complex nonlinear systems,and is widely used in many fields such as physics,biochemistry,electronics,economics,and art and so on.In recent years,great progress has been made in the study of fractional-order chaotic systems.Researchers have proposed a large number of new fractional-order chaotic system models and a variety of synchronization strategies for fractional-order chaotic systems.Due to its complex nature,it has attracted wide attention in the field of control systems and secure communications.However,the related research on fractional-order chaotic systems with special dynamic behavior which is the multi-stable characteristics is still relatively small.And such systems often have more prominent research significance and engineering application value.Therefore,this paper mainly studies the fractional-order chaotic system with multi-stable characteristics.The main work can summarize the following aspects.1.A new self-excited multi-stable fractional-order chaotic system is constructed.Firstly,the stable interval of the fractional-order chaotic system is determined by the fractional stability theorem,so as to select the appropriate fractional order value,and then investigating the multi-stable dynamic properties of the proposed chaotic system by employing conventional nonlinear dynamical analysis tools including equilibrium,phase planes,bifurcation diagrams and Lyapunov exponents,chaos diagrams,etc.Then the Multisim software is used to design the circuit model of the system to carry out the simulation experiment,and the hardware circuit experiment of the chaotic system is also implemented.The experimental results of the circuit are consistent with the corresponding numerical theory analysis,which proves the existence and the correctness of the chaotic system.2.A novel hidden multi-stable fractional-order chaotic system is constructed.Since the fractional-order chaotic system has no equilibrium point,equilibrium point stability analysis is not required.Firstly,the nonlinear dynamics tools are used to research the influence of system parameters on the multi-stationary characteristics of fractional-order chaotic systems.In addition,the system has a special feature which is the offset boosting control,which makes the system more extensive and practical in engineering applications.Finally,the circuit simulations and the hardware experiments of the proposed fractional-order chaotic system are carried out,and the results are consistent with the corresponding theoretical analysis.3.The synchronization problem has always been an important topic in the research of chaos.In the work of this paper,the sliding mode synchronization theory and the finite-time synchronization theory are briefly introduced.Then,according to the above synchronization theory,two suitable fractional-order synchronous controllers are designed and introduced into the two different proposed fractional-order chaotic system models,so as to realize the synchronization of the fractional-order chaotic system.