Frequency-Adaptive Learning Duffing System

• Frequency-adaptive learning Duffing system is a nonlinear oscillator whose natural frequency evolves via a learning rule derived from the excitation input.
• It achieves broadband vibrational resonance amplification as demonstrated through rigorous theory, high-resolution simulations, and validated circuit implementations.
• The system offers robust weak-signal detection and effective denoising in applications like RF tag readings, fault diagnostics, and adaptive control.

A frequency-adaptive learning Duffing system is a nonlinear dynamical device whose instantaneous natural frequency evolves according to a learning rule derived from the excitation input, allowing the system to dynamically retune its resonance properties to match the frequencies present in its driving signals. This approach extends and stabilizes vibrational resonance (VR)—a phenomenon where a system exhibits pronounced amplification in response to a weak signal when also excited by a strong high-frequency auxiliary input—across a broad range of operating frequencies. The methodology and efficacy of such systems have been demonstrated via rigorous theoretical analysis, high-resolution numerical simulations, and validated circuit implementations (Wan et al., 10 Jan 2026, Yang et al., 10 Jan 2026).

1. Mathematical Formulation

The canonical frequency-adaptive learning Duffing system is governed by the coupled dynamical equations: $$\begin{cases} \ddot x + 2\,\zeta\,\dot x + \omega^2(t)\,x + b\,x^3 = \beta \left[ A \cos(\Omega_1 t) + B \cos(\Omega_2 t) \right] \ \dot \omega = k_\omega \left[ A \cos(\Omega_1 t) + B \cos(\Omega_2 t) \right] \end{cases}$$ where $x(t)$ is the oscillator state, $\omega(t)$ is the instantaneous natural frequency, $b$ is the Duffing cubic stiffness (hardening for $b>0$, softening for $b<0$), $\zeta$ the damping ratio, $\beta$ the input amplification, and $k_\omega$ the learning rate controlling frequency adaptation. The inputs $A\cos(\Omega_1 t)$ (weak) and $B\cos(\Omega_2 t)$ (auxiliary, typically $\Omega_2 \gg \Omega_1$) serve both as drive and as the source for frequency learning.

On integrating the frequency learning rule, the explicit solution is: $$\omega(t) = k_\omega \left[ \frac{A}{\Omega_1}\sin(\Omega_1 t) + \frac{B}{\Omega_2} \sin(\Omega_2 t) \right]$$ and the squared frequency, separating slow and fast parts,

$$\omega^2(t) = \frac{k_\omega^2}{2}\left( \frac{A^2}{\Omega_1^2}+\frac{B^2}{\Omega_2^2}\right) - \frac{k_\omega^2 A^2}{2 \Omega_1^2}\cos(2\Omega_1 t) - \frac{k_\omega^2 B^2}{2 \Omega_2^2}\cos(2\Omega_2 t)$$

This time-dependent $\omega^2(t)$ couples parametric and translational excitation, allowing the system to dynamically follow variations in excitation frequency.

2. Resonance Condition and Theoretical Analysis

The resonance enhancement for a weak characteristic frequency $\Omega_1$ occurs when $\omega_{\textrm{eff}} \approx \Omega_1$, i.e., the slow mean of the time-varying frequency matches the input: $$\Omega_1^2 = \frac{k_\omega^2}{2}\left( \frac{A^2}{\Omega_1^2} + \frac{B^2}{\Omega_2^2}\right)$$ Critical parameter values can be extracted: $$B_c = \sqrt{\frac{2\Omega_2^2}{k_\omega^2}\left( \Omega_1^4 - \frac{k_\omega^2 A^2}{2} \right)}$$

$$\Omega_{2c} = \frac{k_\omega B}{\Omega_1} \sqrt{\frac{1}{2}(2\Omega_1^4 - k_\omega^2 A^2)}$$

These first-order estimations provide practical guidance for tuning resonance amplification. The broadening of the VR band is a direct consequence of the adaptive frequency law, which is analytically traced via averaging and confirmed by numerical experiments (Wan et al., 10 Jan 2026, Yang et al., 10 Jan 2026).

3. Simulation and Experimental Validation

Numerical simulations employ explicit Euler integration with fine step ($\Delta t = 0.001$), measuring the steady-state response amplitude $Q$ at $\Omega_1$ using: $Q = \frac{\sqrt{Q_s^2 + Q_c^2}}{A}$ with $Q_s$, $Q_c$ the Fourier components over $mT$ cycles post transient decay. Observed phenomena include:

• $Q$ reaches a maximum at $B = B_c$ and diverges for excessive $B$.
• Parameter sweeps (e.g. $k_\omega$, $\beta$, $b$) reveal nonlinearly shifting $B_c$, optimal $k_\omega$, and robustness to $b<0$.
• Systems with frequency adaptation (learning rule) achieve resonance and stability where fixed-frequency Duffings fail, especially for softening ($b<0$). Experimental circuit-realizations (e.g. Multisim, op-amp and multiplier-based) replicate dynamical equations and validate VR predictions, showing quantitative agreement for $B_c$ and amplification ratios ($\sim8\times$–$10\times$ in hardware for matched theory and simulation) (Wan et al., 10 Jan 2026).

4. Comparison of Learning Rules

Table: Learning Rule Properties

Rule	Stability Bandwidth	Peak Resonance $Q_{max}$	Divergence Behavior
$\dot\omega = k_\omega f(t)$	Broad	Largest	Minimal divergence
$\dot\omega = k_\omega x f(t)$	Narrow	Lower	Frequent divergence
$\dot\omega = k_\omega \dot x f(t)$	Narrow	Lower	Frequent divergence

The simple adaptive rule ($\dot\omega = k_\omega f(t)$) outperforms Hebbian alternatives both in stability and resonance degree. Rules coupling explicitly with state or derivative induce higher instability and narrower operating windows (Wan et al., 10 Jan 2026).

5. Coupled Arrays and Signal Processing Applications

Arrays of $n$ unidirectionally coupled adaptive Duffing oscillators are governed by: $$\begin{aligned} &\ddot{x}_i + 2\zeta\dot{x}_i + \omega_i^2(t)x_i + b x_i^3 = \beta \left[ f(t) + \varepsilon x_{i-1} \right] \ &\dot{\omega}_i = k_\omega \left[ f(t) + \varepsilon x_{i-1} \right] \end{aligned}$$ where $f(t) = A\cos(\Omega_1 t) + B\cos(\Omega_2 t)$, $\varepsilon$ is coupling strength. Each oscillator tracks the dominant frequency component in its input, and cascaded coupling amplifies the VR effect. When driven by noisy signals (AWGN SNR $-10\,$dB to $+10\,$dB), the VR array progressively denoises and amplifies the weak component, outperforming conventional methods (wavelet denoising, Kalman filtering) especially under multipath or strong noise conditions. Experimental radio-frequency (RF) tag readings and fault diagnostics confirm superior extraction of characteristic frequencies (Yang et al., 10 Jan 2026).

6. Engineering Implementation and Limitations

Analog circuit instantiation is achieved using resistors, capacitors, multipliers, and gain stages in modular topologies. Learning law implementation is feasible via integrator circuits or digital filtering, and programmable gain amplifiers realize input scaling ($\beta$). Moderate coupling ($\varepsilon \sim 1$) is optimal for robust VR amplification and avoidance of divergences.

Fundamental limitations remain: analytical treatment is restricted to mean-field averaging, and full stability boundaries (especially in $(b, k_\omega)$ space) are open questions. Adaptation presently targets primary harmonic content; generalized learning for unknown or broadband inputs, stochastic extension, and real-time VLSI circuits are active research fronts (Wan et al., 10 Jan 2026, Yang et al., 10 Jan 2026).

7. Related Adaptive Control Frameworks

Machine learning approaches, particularly deep regression pipelines for real-time parameter tuning as described in the context of Lyapunov-controlled chaotic Duffing–Van der Pol systems, offer alternative mechanisms for frequency adaptation. Here, model parameters (including excitation frequency $\omega$) are adaptively re-tuned by a deep neural network to maintain exponential stability, minimize tracking error, and rapidly counteract system disturbances. This suggests a convergence between principled learning rules and data-driven adaptive control for nonlinear, chaotic systems, with real-world demonstrations on rapid recovery times and enhanced robustness (Mahmoud et al., 2020).

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Collectively, frequency-adaptive learning Duffing systems constitute a rigorously analyzed, experimentally validated methodology for broad-band vibrational resonance amplification, stable weak-signal detection, and robust denoising. The paradigm leverages both analytical derivations and modern machine learning control frameworks, offering significant utility for nonlinear system optimization and signal-processing applications.