In this paper, a novel passive memristor model and its equivalent circuit model are designed, analysed and realised to investigate the memristor characteristics and their applications in nonlinear circuits. By employing this memristor model, a new third-order memristive Wien bridge is set up. Dynamical behaviours of the system are studied in detail, and multiscroll attractors, coexisting bifurcation modes, coexisting attractors, antimonotonicity and transient chaotic bursting are observed in this system by using theoretical analysis, simulation analysis and circuit experiment. Integrable deformation of the memristive Wien-bridge system is analysed. The circuit experiment is performed by replacing the memristor with its equivalent circuit model in the proposed memristor-based Wien-bridge circuit.

Meminductor is a novel nonlinear inductor with memory. A meminductor-based chaotic oscillating circuit that has only two linear resistors, two linear capacitors and a meminductor, is designed based on a mathematical model of the flux-controlled meminductor to study its characteristics in nonlinear circuit. Through the analysis of bifurcations, dynamic map and Lyapunov exponents, it is found that the system can exhibit some complex characteristics, such as an infinite number of equilibrium points and burst chaos. Especially, bifurcation without parameters and coexisting attractors appear under a fixed set of parameter values but different initial conditions. Moreover, random characteristics of the PN sequences generated from the chaotic circuit are tested via the test suit of National Institute of Standards and Technology (NIST), and the tested randomness definitely reaches the standards of NIST. Finally, a scheme for digitally realising this oscillating circuit is provided using the digital signal processor (DSP).

In this paper, a fractional-order locally active memristor is proposed based on the definition of fractional derivative. It is found that the side lobe area of the pinched hysteretic curve of the memristor changes with the fractional-order value, and the side lobe’s area of the fractional-order memristor is greater than that of them emristor with integer order, meaning that the memory of the fractional-order memristor is stronger than that of the memristor with integer order. It is proved by the dynamic rout map (DRM) that the fractional-order memristor possesses continuous memory. The pinched hysteresis, memristance and power characteristics which vary with the fractional order are compared and analysed in detail. Furthermore, we use the memristor to construct a fractional-order chaotic circuit, which can exhibit continuous chaotic motion in the range of 0.75 < fractional order α < 1 and various coexisting attractors. We also show that the lower fractional order causes higher complexity of the fractional order chaotic system using different methods, such as Lyapunov exponent spectrum, bifurcation diagram, spectral entropy and C0 complexity method. Finally, the circuit simulations of the fractional-order chaotic circuit are realised, demonstrating the validity of the mathematical model and the theoretical analysis.