Investigation of chaos and memory effects in the Bonhoeffer-van der Pol oscillator with a non-ideal capacitor

Abstract

In this paper, the voltage fluctuations of the Bonhoeffer van der Pol oscillator system with a non-ideal capacitor were investigated. Here, the capacitor was modeled, using a fractional differential equation in which the order of the fractional derivative, α, is also a measure of the memory in the dielectric. The governing fractional differential equation was derived using two methods, namely a differential and integral approach. The former method utilized a hierarchical resistor-capacitor (RC) ladder model while the latter utilized the theory of the universal dielectric-response. It was found that the dynamical behavior of the potential across the capacitor was affected by the parameter α and, therefore, the memory of the system. Additionally, findings indicate that an increase in the memory parameter was associated with an increase in the energy stored in the dielectric. Furthermore, the effects of the dynamical behavior of the voltage on the capacity of the dielectric to store energy was examined. It was found that oscillation death resulted in a higher amount of stored energy in the dielectric over time, as compared to behavior, which displayed relaxation oscillations or chaotic fluctuations. The relatively-lower stored energy resulting from the latter types of dynamical behavior appeared to be a consequence of the memory effect, where the present accumulation of energy in the capacitor is reduced by previous decreases in the potential. Hence, in this type of scenario, the dielectric material can be thought of as “remembering” the past behavior of the voltage, which leads to either a decrease, or an enhancement in the stored energy. Moreover, an increase in the fractional parameter α, under certain conditions, led to the earlier onset of the chaotic voltage oscillations across the capacitor. On the other hand, the corresponding phase portraits showed that the chaotic behavior was heightened, in general, with a decrease in α. The non-ideal capacitor was also found to have a transitory nature, where it behaves more like a resistor as α → 0, and conversely, more like a capacitor as α → 1. Here, a decrease in α was linked to an enhanced metallic character of the dielectric. Finally, a possible link between the complexity of the voltage noise fluctuations and the metallic character of the non-ideal capacitor will be discussed.