Extreme multistability has frequently been reported in autonomous circuits involving memory-circuit elements, since these circuits possess line/plane equilibrium sets. However, this special phenomenon has rarely been discovered in non-autonomous circuits. Luckily, extreme multistability is found in a simple non-autonomous memcapacitive oscillator in this paper. The oscillator only contains a memcapacitor, a linear resistor, a linear inductor, and a sinusoidal voltage source, which are connected in series. The memcapacitive system model is firstly built for further study. The equilibrium points of the memcapacitive system evolve between a no equilibrium point and a line equilibrium set with the change in time. This gives rise to the emergence of extreme multistability, but the forming mechanism is not clear. Thus, the incremental integral method is employed to reconstruct the memcapacitive system. In the newly reconstructed system, the number and stability of the equilibrium points have complex time-varying characteristics due to the presence of fold bifurcation. Furthermore, the forming mechanism of the extreme multistability is further explained. Note that the initial conditions of the original memcapacitive system are mapped onto the controlling parameters of the newly reconstructed system. This makes it possible to achieve precise control of the extreme multistability. Furthermore, an analog circuit is designed for the reconstructed system, and then PSIM circuit simulations are performed to verify the numerical results.
Extreme multistability usually emerges in a mem-element’s circuit or system that possesses a line or plane equilibrium set closely associated with the internal initial state of the mem-element. To extend the investigation of extreme multistability, this paper proposes a nonautonomous memcapacitive oscillator, discovering a new type of extreme multistability due to the infinitely many discrete equilibrium points therein. This memcapacitive oscillator is constructed by connecting a simple memcapacitor-resistor circuit with a sinusoidal voltage. With its normalized model, the infinitely many discrete equilibrium points are computed and the infinitely many necklace-shaped coexisting attractors that were not yet reported are disclosed by numerical methods. Since the number and stability of the equilibrium points vary with time, the attraction basins with complex ripple structures are formed in the memcapacitive oscillator, resulting in the appearance of a special type of extreme multistability. Furthermore, PSIM circuit simulations and microcontroller-based hardware experiments are performed to verify the numerical results.
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