| The appearance of the memristor circuit element not only makes up the lack of traditional circuit system theory,but also memristor together with the traditional basic element such as resistance,inductance and capacitance,constitutes complete circuit theoretical basis.At the same time,when the current passing through the memristor changes,it can cause the unique physical structure property of the resistance change,which makes memristor become a non-volatile information memory with superior performance,so this feature leads to great potential in the application of memristor.In the research process of chaos system and circuit simulation,the simulation process of nonlinear system or the mathematical model of chaos system can be replaced the state variables of chaos system by the nonlinear circuit model of memristor.This process can reduce the cost of simulation research and development,adjust the system parameters directly by adjusting the circuit parameters,or adjust the system parameters by building a nonlinear circuit model to verify the theoretical analysis results of the chaotic system,the simulation model of the complex system is transformed into a more convenient way to study the nonlinear circuit system.In this paper,a new hyperchaotic system is designed for Lü chaotic system and charge controlled memristor,and its dynamic behavior,chaos control and chaos synchronization are analyzed.Finally,the application of hyperchaotic system in chaotic secure communication system is studied by using chaos masking scheme.The main contents are as follows.On the basis of Lü system,this paper introduces the mathematical model of its three-dimensional chaotic system and analyzes its dissipation,symmetry in phase space and bifurcation characteristics.According to the distribution of the eigenvalues corresponding to the equilibrium point of the chaotic system in the complex plane,the type of the equilibrium point can be determined to judge the chaotic characteristics of the system at the equilibrium point;the differential equations of the mathematical model of the hyperchaotic system can be solved by using the fourth-order Runge-Kutta method to study the motion state of the system,and the time history diagram and phase trajectory diagram of each state variable of the system in the time domain can be calculated bifurcation diagram,Poincare map and Lyapunov exponent spectrum.Based on the mathematical model of the memristor circuit element in the three-dimensional Lü chaotic system,the physical model of the memristor electronic element is transformed into a function term and used as a feedback term.The feedback term is added to the chaotic system to design a new four-dimensional hyperchaotic system model.The hyperchaotic system is taken as the research object,combining the bifurcation theory,fractional stability theory and Lyapunov function stability theory.In this paper,the stability ofthe integer order and fractional order models of the new hyperchaotic system affected by the system parameters is studied theoretically.The fourth-order Runge-Kutta method is used to simulate the integer order hyperchaotic mathematical model,and the prediction correction algorithm is used to solve the fractional order system model of the above-mentioned hyperchaotic system,and the stability region of the new four-dimensional hyperchaotic system with the order change is calculated.The dynamic behavior of the system is analyzed and the theoretical results are proved by using the phase trajectory and time history of the hyperchaotic system.It is found that the system is very sensitive to the change of system parameters in the process of time iteration.According to the adaptive control theory,one-way coupling feedback control theory,fractional order control theory and so on,we try to find out the control method which can be applied to the hyperchaotic system and has fast control speed,and through the construction of Lyapunov function and other methods,we analyze the stability of the hyperchaotic system under the controlled condition,and conclude that the controlled system is stable in the time domain,and then to the controlled system.The simulation results show that the state variable curve of the error system is stable at zero point,and then it can be proved that the controller can control the hyperchaotic behavior of the memristor hyperchaotic system very well.According to the new hyperchaotic system based on memristor designed in this paper,the drive response system is obtained by adding synchronization controller on the basis of its mathematical model,then the error system model corresponding to the hyperchaotic synchronization system is established by making the difference between the response system and the drive system,and the synchronization problem of each corresponding state variable of the hyperchaotic system is transformed into the corresponding error of the error system.The stability of the variables is analyzed,and the validity of the synchronous control can be deduced by proving the stability of each variable of the error system in the time domain.Based on the theory of fractional order stability,the distribution of eigenvalues of Jacobian matrix of the synchronization error system of hyperchaos is studied.The distribution of eigenvalues and the main value of eigenvalues are calculated to determine the stability interval of the corresponding fractional order.The calculation results show that the synchronization error system of hyperchaos is stable under the condition of a certain fractional order.The synchronization error system will be obtained.When the system is stable,the fractional order is combined with the synchronization controller to realize the synchronization of the chaotic behavior of the fractional hyperchaotic system.Due to the high complexity and other chaotic signal characteristics of the chaotic signal generated by the hyperchaotic system with memristor,the chaotic signal of the hyperchaotic system is particularly suitable for the requirements of the information transmission carrier ofthe chaotic secure communication system.In this paper,the problem of chaotic secure communication based on the new memristor hyperchaotic system is studied.Based on the scheme of chaos masking in chaos secure communication theory,the synchronization system of hyperchaos system under generalized synchronization is designed.By coupling any state variable in the synchronous drive system with the transmitted signal,a complex mixed signal with similar chaotic signal and noise characteristics is formed,and the mixed signal is transformed into the corresponding state variable waveform of the response system.After the signal is separated,the useful signal can be transmitted in the hyperchaotic synchronization system,so that the signal can be transmitted by the hyperchaotic synchronization system. |
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