Vibrational resonance in a frequency-adaptive learning Duffing system

TL;DR

This work addresses the limited frequency range of vibrational resonance in conventional Duffing oscillators by introducing a frequency-adaptive learning rule that dynamically tunes the natural frequency in response to external excitations. Through numerical simulations, approximate theoretical insights, and circuit simulations, the authors demonstrate that the adaptive system achieves stronger and more robust vibrational resonance across a broader range of frequencies than the traditional setup. Key findings include the beneficial impact of the learning rule on stability, a comparative advantage over Hebbian-like alternatives, and practical validation via a Multisim circuit. The results offer design guidelines for applying nonlinear resonators to weak-signal processing tasks in engineering contexts.

Abstract

Vibrational resonance focuses on the resonance behavior of a nonlinear system when it is subjected to both a weak low-frequency characteristic signal and a high-frequency auxiliary signal. A traditional Duffing system has a fixed natural frequency and lacks adaptability to the excitation frequency, resulting in vibrational resonance occurring only in a lower frequency range, which affects the application of vibrational resonance. We propose a frequency-adaptive learning Duffing system to overcome the above problem through a learning rule of the natural frequency. The optimal vibrational resonance performance is demonstrated by examining the influence of auxiliary signal parameters, nonlinear stiffness coefficient and the learning rule on the response. The appearance of vibrational resonance is verified by numerical simulation, approximated theoretical predication and circuit simulation. In addition, the advantages of the proposed frequency-adaptive learning rule are highlighted in vibrational resonance performance by comparing with that of two other commonly used alternatives called Hebbian learning rules. The proposed learning rule makes the system more stable and have a stronger resonance degree. The results provide a useful reference for optimizing nonlinear system response and also for processing a weak characteristic signal through nonlinear resonance methods. These achievements provide a groundbreaking foundation for future applied studies especially in the field of weak and complex signal processing.

Figures (14)

• Figure 1: The response amplitude $Q$ versus the high-frequency signal amplitude $B$ and the nonlinear stiffness coefficient $b$ calculated from Eq. (1). (a) There is not a bona fide vibrational resonance due to the small value of $Q$ when $b>0$. (b) There is not vibrational resonance appearing and the response diverges as $B$ increasing when $b<0$. The simulation parameters are $\zeta=0.1$, $\omega_0=1$, $A=0.1$, $\Omega_1=5$ and $\Omega_2=50$.
• Figure 2: The response amplitude $Q$ versus the high-frequency signal amplitude $B$ and the nonlinear stiffness coefficient $b$ calculated from Eq. (5). (a) Vibrational resonance appears when $b>0$ indicating the necessity of the amplification $\beta$. (b) There is not vibrational resonance appearing and the response diverges as $B$ increasing when $b<0$. The simulation parameters are $\zeta=0.1$, $\omega_0=1$, $\beta=10$, $A=0.1$, $\Omega_1=5$ and $\Omega_2=50$.
• Figure 3: Vibrational resonance induced by the auxiliary signal. (a) The response amplitude $Q$ presents obvious vibrational resonance band in the $B-\Omega_2$ plane. (b) The maximal value of the response amplitude $Q_{max}$ versus the corresponding critical value $B_c$. (c) The critical values of $B_c$ versus the high-frequency $\Omega_2$ are obtained by numerical simulation and approximated theoretical predication in Eq. (11) respectively. (d) The maximal value of the response amplitude $Q_{max}$ versus the high-frequency $\Omega_2$. The simulation parameters are $\zeta=0.1$, $b=1$, $k_\omega=10$, $\beta=10$, $A=0.1$ and $\Omega_1=5$.
• Figure 4: Effect of the amplification factor $\beta$ on vibrational resonance that induced by the auxiliary signal. (a) The response amplitude $Q$ presents obvious vibrational resonance band in the $B-\beta$ plane. (b) The maximal value of the response amplitude $Q_{max}$ versus the corresponding critical value $B_c$. (c) Critical value $B_c$ versus the excitation amplification factor $\beta$. (d) The maximal value of the response amplitude $Q_{max}$ versus the amplification factor $\beta$. The simulation parameters are $\zeta=0.1$, $b=1$, $k_\omega=10$, $A=0.1$, $\Omega_1=5$ and $\Omega_2=50$.
• Figure 5: Effect of the characteristic frequency $\Omega_1$ on vibrational resonance that induced by the auxiliary signal. (a) The response amplitude $Q$ presents obvious vibrational resonance band in the $B-\Omega_1$ plane. (b) The maximal value of the response amplitude $Q_{max}$ versus the corresponding critical value $B_c$. (c) The critical values of $B_c$ versus the characteristic frequency $\Omega_1$ are obtained by numerical simulation and approximated theoretical predication in Eq. (11) respectively. (d) The maximal value of the response amplitude $Q_{max}$ versus the signal frequency $\Omega_1$. The simulation parameters are $\zeta=0.1$, $b=1$, $k_\omega=10$, $\beta=10$, $A=0.1$ and $\Omega_2 = 10\Omega_1$.