Extreme Multistability and Its Incremental Integral Reconstruction in a Non-Autonomous Memcapacitive Oscillator

Abstract

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.

Chaos Synchronization of Two Györgyi–Field Systems for the Belousov–Zhabotinsky Chemical Reaction

“…In systems with multi-metastable stages, the phase trajectory of the attractors depends on the initial conditions [31]. For specific conditions at low flux (Table 1), the bidimensional attractors are open cycles where the portraits of the phases do not overlap (Figures 4-6), which confirms the hypothesis that the system has a chaotic behavior.…”

Quality Evaluation for Reconstructing Chaotic Attractors

“…One can say that a dynamic system is called chaotic if solutions are found in a permanently bordered area B ⊂ n of the phase space and have the following fundamental characteristics: • Fourier transform (power spectrum) of any of the state variables is similar to white noise. This property indicates the appearance of a non-periodic chaotic trajectory [22].…”

Dynamics of a Reduced System Connected to the Investigation of an Infinite Network of Identical Theta Neurons

“…Considering a small network of identical all-to-all coupled phase oscillators, chaotic behaviour was found in networks of four or more phase oscillators with a single harmonic in their coupling function [11,12]. Many other types of neuron models and their firing activities have been studied in papers such as [13][14][15][16], where, specifically, the dynamics of a discrete memristive Rulkov neuron model which uses a memristor to describe the magnetic induction effects of the neuron and, respectively, a single neuron model with memristive synaptic weight, have been investigated.…”