Dynamic Analysis Of Nonlinear Circuit With Fractional-order Memristor

In 1971,Cai Shaotang,a Chinese professor at the University of California,Berkeley,proposed the memristor model for the first time,which linked the two variables of charge and magnetic flux in theory to make up for the lack of the relationship between these two basic circuit variables.In 2008,a research team at HP Labs confirmed the existence of a memristor model in the laboratory.The non-linear circuit built with the memristor has complex dynamic characteristics and is worthy of further research.First,the basic theory of fractional-order and common memristor models are introduced.The robust stability of a fractional-order linear time-invariant system is analyzed.A system with two different fractional-order uncertainty intervals and a parameter uncertainty interval is studied.The distribution of the stability region in the complex plane is analyzed.Then a nonlinear circuit model with two fractional-order cubic smooth memristors is designed,and the stability of the fractional-order system under specific parameters is analyzed.It is found that the change of the order has little effect on the stability of the system.The influence of the circuit parameter changes on the system under different orders is studied,and the interval distribution of unstable parameters of the system under different orders is obtained.At the same time,the transient chaos and state transition in the circuit are analyzed.Finally,non-linear conductance is introduced into the integer-order non-smooth memristor circuit.The stability of the system is analyzed from two aspects of mathematical analysis and numerical simulation,and the Hopf bifurcation and saddle-node bifurcation appear when the system stability changes.The phenomenon of "intermittent chaos" and the period doubling bifurcation in non-smooth nonlinear systems are found.The bifurcation diagram and Lyapunov exponential spectrum are also used to study the stability of the system when the capacitance and resistance were changed,and a wealth of chaotic dynamic behavior is obtained.