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A new hyperbolic-type memristor emulator is presented and its frequency-dependent pinched hysteresis loops are analyzed by numerical simulations and confirmed by hardware experiments. Based on the emulator, a novel hyperbolic-type memristor based 3-neuron Hopfield neural network (HNN) is proposed, which is achieved through substituting one coupling-connection weight with a memristive synaptic weight. It is numerically shown that the memristive HNN has a dynamical transition from chaotic, to periodic, and further to stable point behaviors with the variations of the memristor inner parameter, implying the stabilization effect of the hyperbolic-type memristor on the chaotic HNN. Of particular interest, it should be highly stressed that for different memristor inner parameters, different coexisting behaviors of asymmetric attractors are emerged under different initial conditions, leading to the existence of multistable oscillation states in the memristive HNN. Furthermore, by using commercial discrete components, a nonlinear circuit is designed and PSPICE circuit simulations and hardware experiments are performed. The results simulated and captured from the realization circuit are consistent with numerical simulations, which well verify the facticity of coexisting asymmetric attractors' behaviors.

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By replacing the Chua's diode in Chua's circuit with a first-order hybrid diode circuit, a fourthorder modified Chua's circuit is presented. The circuit has an unstable zero saddle point and two nonzero saddle-foci. By Routh-Hurwitz criterion, it is found that in a narrow parameter region, the two nonzero saddle-foci have a transition from unstable to stable saddle-foci, leading to generations of self-excited and hidden attractors in the modified Chua's circuit simultaneously, which have not been previously reported. Complex dynamical behaviors are investigated both numerically and experimentally. The results indicate that the proposed circuit exhibits complicated nonlinear phenomena including self-excited attractors, coexisting self-excited attractors, hidden attractors, and coexisting hidden attractors.

In order to overcome the boundary effect and boundary lock problem existing in classical Hewlett-Packard (HP) TiO 2 non-linear model, the authors propose a novel window function for the fractional-order HP TiO 2 non-linear drift model, in which the fractional calculus is utilised to reflect the memory property of the memristor device. The novel window function is general and they can take the previously reported well-known window functions as its special cases by turning parameter a. Compared with the integer-order model, the order α and a in the fractional-order case is important parameters to flexibly realise the nonlinear dopant drift of memristor model even when a wider amplitude range of the input voltage is applied. Simulation results illustrate that their model is flexible, scalable to guarantee the state variable x(t) and the memristor value M α (x) switched between the low and high levels by choosing suitable parameter α and a. A simple practical application also confirms the efficiency of their model to reveal the non-linear dopant kinetics of the memristor device.