Fractioanl-order Nonlinear Dynamical Synchronization And Nonlinear Circuit

As we know, the biological system is nonlinear in nature, and there are full of the complex nonlinear systems, which can’t be replaced by the linear systems. As for the chaos, it’s an important variety of the nonlinear dynamics, which reveals that the complexity, determinacy and randamicity, therefore, it’s a better way to study the chaos for knowing the nonlinear nature of the biological system. In recent years, many research findings revealed that the mathematical model based on the fractional calculus can show the electrical characteristics better. Therefore, it’s an important issure to control the chaos for system stability, system security, system capability, which leads the develotment of biological system to meet peoples’ wishes. So, we study the synchronization between the integer order chaos systems and the fractional-oeder chaos systems from the point of the nonlinear dynamics theories. And we also discuss the fractional-order nonlinear circuits theories and their applications and the controllability of the fractional-order complex circuits networks.The contents and conclusions of this paper include:(1) Chaos exists in the biological system. As for chaos synchronization, we realized the the synchronization between the integer order chaos systems and the fractional-oeder chaos systems by designing a slipping model and choosing the proper controlling parameters, which indicated the availability of the method. Second, in order to bridge between fractional-order and integer order nonlinear dynamical system, in this letter, we brings attention to synchronization and anti-synchronization between fractional-order chaotic system and integer-order chaotic system by using back-stepping method. And the sufficient conditions for achieving the synchronization and anti-synchronization of a class of fractional-order nonlinear system and integer-order nonlinear system are derived based on Lyapunov stability theory. Additionally, numerical simulations are provided to verify the effectiveness and feasibility of the proposed control scheme, which are in agreement with theoretical analysis. Finally, we present a general form of a class of chaotic system, which can besynchronized between a fractional-order chaotic system and an integer order chaotic system. Furthermore, an example is carried out to verify and demonstrate the effectiveness of the proposed control scheme. Simultaneously, our work is supported by logical theorems and intuitive numerical simulation, which provides the theoretical basis for the synchronization of the biological systems.(2)The constructions of the biological system can’t do without the circuit theory. As for the fractional-order nonlinear circuits’ theories, we studied the electrical characteristics of the fractional-order RLC series circuit and the fractional-order Foster II LC circuit. Specifically, we study the effects of frequency, circuit parameters R, L, C and two new parameters, α and β on two regular indicators: magnitude resistance ratio and phase response. Moreover, theoretical analysis and numerical simulations are both proposed. As a peculiar phenomena-resonance, the relationship among resonance frequency, fractional order and LC is studied in detail. A graphic description of the changing trend of resonance frequency versus α-β by using 3d drawing is presented for the first time. In addition, sensitivity analysis including some interesting rules is illustrated. Finally, to promote industrial application, typical examples based on the above-mentioned results are prior presented. Importantly, the study results showed that the fractional-order nonlinear circuits had more flexibility and selectivity, and it can help know the complex principles of biologies.(3)The study and construction of biological system rely on the nonlinear circuit. As for the fractional-order filter circuits, this part introduces the fundamentals of the conventional RLC filter circuit in the fractional domain. First, the impedance characteristics and phase characteristics of the RLC filter circuit are studied. Moreover, from the filtering property perspective, we systematically investigate the effects of the system variables on the amplitude-frequency characteristics and phase-frequency characteristics. In addition, the detailed analyses of the cut-off frequency are presented. And the study results show that the fractional-order filter circuits have the better filter prosperity, which provides the technical support for the study of biological system.(4)The complex circuit networks can describe the complex characteristics of the biological systems properly. As for the controllability of the fractional-order nonlinear complex circuits networks, the mathematical expression and prove of the controllability of the fractional-order nonlinear complex circuits networks are presented. Furthermore, the controllability condition of the fractional-order RLC series circuit and the fractional-order RLC paralleling circuit are studied. Simultaneously, our work is supported by logical theorems and intuitive numerical simulation. Additionally, the mathematical expression and prove of the controllability of the fractional-order cascaded complex circuits networks are also presented. Similarly, the numerical simulatioan results are in good agreement with theroretical analysis. And the realization of the controllability of the complex circuit networks provides a new pantform where the realization of the controllability of the biological systems can be realized.