Transient transition behaviors of fractional-order simplest chaotic circuit with bi-stable locally-active memristor and its ARM-based implementation

This paper proposes a fractional-order simplest chaotic system using a bi-stable locally-active memristor.The characteristics of the memristor and transient transition behaviors of the proposed system are analyzed,and this circuit is implemented digitally using ARM-based MCU.Firstly,the mathematical model of the memristor is designed,which is nonvolatile,locally-active and bi-stable.Secondly,the asymptotical stability of the fractional-order memristive chaotic system is investigated and some sufficient conditions of the stability are obtained.Thirdly,complex dynamics of the novel system are analyzed using phase diagram,Lyapunov exponential spectrum,bifurcation diagram,basin of attractor,and coexisting bifurcation,coexisting attractors are observed.All of these results indicate that this simple system contains the abundant dynamic characteristics.Moreover,transient transition behaviors of the system are analyzed,and it is found that the behaviors of transient chaotic and transient period transition alternately occur.Finally,the hardware implementation of the fractional-order bi-stable locally-active memristive chaotic system using ARM-based STM32F750 is carried out to verify the numerical simulation results.

Firing activities in a fractional-order Hindmarsh–Rose neuron with multistable memristor as autapse

Considering the fact that memristors have the characteristics similar to biological synapses, a fractional-order multistable memristor is proposed in this paper. It is verified that the fractional-order memristor has multiple local active regions and multiple stable hysteresis loops, and the influence of fractional-order on its nonvolatility is also revealed. Then by considering the fractional-order memristor as an autapse of Hindmarsh–Rose(HR) neuron model, a fractional-order memristive neuron model is developed. The effects of the initial value, external excitation current, coupling strength and fractional-order on the firing behavior are discussed by time series, phase diagram, Lyapunov exponent and inter spike interval(ISI) bifurcation diagram. Three coexisting firing patterns, including irregular asymptotically periodic(A-periodic)bursting, A-periodic bursting and chaotic bursting, dependent on the memristor initial values, are observed. It is also revealed that the fractional-order can not only induce the transition of firing patterns, but also change the firing frequency of the neuron. Finally, a neuron circuit with variable fractional-order is designed to verify the numerical simulations.