On the Study of Chaos and Memory Effects in the Bonhoeffer-van der Pol Oscillator with a Non-Ideal Capacitor

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 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. The dynamical behavior of the potential across the capacitor was found to be affected by this parameter, 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. 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 present accumulations of energy in the capacitor are affected 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. The non-ideal capacitor was also found to have a transitory nature, where it behaves more like a resistor as {\alpha} approaches 0, and conversely, more like a capacitor as {\alpha} goes to 1. Here, a decrease in {\alpha} was linked to an enhanced metallic character of the dielectric.

💡 Research Summary

The paper investigates how a non‑ideal capacitor, modeled with a fractional differential equation, influences the dynamics of a forced Bonhoeffer‑van der Pol (BVP) oscillator. The fractional order α (0 < α ≤ 1) serves as a quantitative measure of dielectric memory: α ≈ 1 corresponds to an ideal capacitor, while α ≈ 0 reduces the element to a pure resistor. Two independent derivations are presented. The first uses a hierarchical RC‑ladder network; applying Kirchhoff’s laws, Laplace transforms, and continued‑fraction expansions shows that an infinite ladder converges to an impedance proportional to s^‑α, i.e., a fractional capacitor. The second follows the universal dielectric response, expressing polarization as a convolution of a power‑law kernel f(t) ∝ t^‑α with the electric field; differentiating yields i(t) = C_α d^αv(t)/dt^α. Both approaches produce the same governing fractional differential equation for the capacitor current.

Numerical simulations replace the ordinary capacitor in the standard BVP circuit with the fractional element while keeping the external forcing V₁ sin(ωt) and DC bias V₀ fixed. By varying α (e.g., 0.6, 0.8, 1.0) the authors examine time series of the capacitor voltage θ(t), phase‑space trajectories (θ, dθ/dt), and the average stored energy ⟨E⟩ = ½ C_eff⟨θ²⟩. The results reveal several key phenomena. First, increasing α advances the onset of chaotic oscillations: for a given forcing amplitude and frequency, chaotic behavior appears at lower α values, whereas smaller α values tend to suppress chaos and favor regular relaxation oscillations. This indicates that stronger memory (smaller α) stabilizes the system by damping the nonlinear feedback that drives chaos.

Second, the type of dynamical regime strongly affects energy storage. In the “oscillation death” regime, the voltage settles to a fixed point, allowing charge to accumulate continuously; consequently ⟨E⟩ becomes significantly larger than in chaotic or relaxation‑oscillation regimes. In the latter cases, the voltage fluctuates rapidly, and the memory effect causes past voltage drops to reduce the instantaneous current, leading to higher dissipation and lower stored energy.

Third, phase‑space portraits become increasingly intricate as α decreases. The trajectories develop fractal‑like structures and the power spectrum broadens with enhanced high‑frequency components, reflecting the long‑range temporal correlations introduced by the fractional derivative.

The authors also discuss practical material parameters: SiOₓ dielectrics exhibit α ≈ 0.6, while polypropylene approaches α ≈ 1, confirming that the model captures real‑world dielectric behavior. They note that as α → 0 the element behaves resistively, and as α → 1 it recovers the classic capacitor response, illustrating a continuous transition between metallic and insulating character.

From a broader perspective, the fractional order α emerges as a control knob that simultaneously tunes chaos, energy storage efficiency, and voltage noise complexity. This insight suggests new design strategies for memory‑capacitance devices, such as memcapacitors, ferroelectric or perovskite dielectrics, and neuromorphic circuits where tailored memory effects could be exploited to achieve desired dynamical performance.

In conclusion, the study provides a rigorous theoretical foundation for incorporating fractional‑order memory elements into nonlinear oscillators, demonstrates their impact on chaotic dynamics and energy storage, and opens avenues for engineering advanced electronic components with tunable memory characteristics. Future work should focus on experimental validation, extension to larger networks of fractional elements, and real‑time control schemes that leverage the α‑dependent dynamics.