Wavefront propagation in a bistable dual-delayed-feedback oscillator: analogy to networks with nonlocal interactions

TL;DR

The paper addresses how wavefront propagation in bistable media can be controlled by nonlocal interactions and proposes that a single bistable oscillator with two delayed feedback loops can emulate networks with nonlocal coupling. It analyzes the system via the equation $\frac{dx}{dt} = -x(x-a)(x+b) + F(t) + \frac{\gamma}{2}\left(x(t-\tau_1)+x(t-\tau_2)-2x(t)\right)$, with $\tau_1$ fixed and $\tau_2$ varied, using both numerical simulations and an electronic analog prototype. The key findings show that introducing a second delay speeds up deterministic wavefront propagation and lowers the noise threshold for noise-sustained stabilization in stochastic settings, while also preventing noise-induced destruction of fronts; these effects mirror those observed in networks with nonlocal coupling. This work provides a compact hardware surrogate for investigating nonlocal interactions, with potential implications for reservoir computing and Ising-machine-like optimization.

Abstract

In the present research, a bistable delayed-feedback oscillator with two delayed-feedback loops is shown to replicate a network of bistable nodes with nonlocal coupling. It is demonstrated that all the aspects of the nonlocal interaction impact on wavefront propagation identified in networks of bistable elements are entirely reproduced in the dynamics of a single oscillator with two delays. In particular, adding the second delayed-feedback loop allows speeding up both deterministic and stochastic wavefront propagation, achieving stabilization of propagating fronts at lower noise intensity and preventing fronts from noise-induced destruction occurring in the presence of single delayed-feedback. All the revealed effects are studied in numerical simulations and confirmed in physical experiments, showing an excellent correspondence.

Figures (5)

• Figure 1: Schematic illustration of a delayed-feedback oscillator subject to single (panel (a)) and dual (panel (b)) time-delayed feedback.
• Figure 2: (a) Circuit diagram of the experimental setup (Eqs. (\ref{['experimental_model']})). Output voltage $x_{\tau}$ represents a sum of the delayed signals: $x_{\tau}=(x(t-\tau_1)+x(t-\tau_2))/2$. Operation amplifiers are TL072CP; (b) Methodology for estimation of the front propagation velocity $v$ by using space-time plots. The upper inset shows the propagating fronts in quasi-space $\sigma$ at time moment $n=n_*$.
• Figure 3: Deterministic wavefront propagation control in model (\ref{['numerical_model']}) and experimental setup (\ref{['experimental_model']}) thought varying $\tau_2$: (a) Dependence of the normalized wavefront propagation velocity on delay time $\tau_2$ registered in numerical simulations, $v_{\text{sim}}$, and electronic experiments, $v_{\text{exp}}$. Space-time diagrams in panels (b)-(e) illustrate the dynamics in points 1-2 in numerical and experimental dependencies on panel (a). Parameters are: $a=0.5$, $b=0.45$, $\gamma=0.2$, $\tau_1=1000$ (simulations) and $\tau_1=69.5$ ms (experiments). The quasi-space length is $\eta=1005.95$ (simulations) and $\eta=69.8412$ ms (experiments).
• Figure 4: Multiplicative-noise-based wavefront propagation control: Dependencies of the mean wavefront propagation velocity on the variance of parametric noise ($b=0.45 + \xi(t)$) registered in electronic setup (\ref{['experimental_model']}) (panel (a)) and numerical model (\ref{['numerical_model']}) (panel (b)) at $\tau_2=\tau_1$ and $\tau_2<\tau_1$. The upper inset in panel (a) shows the experimentally obtained dependencies in ranges $<v_{\text{exp}}>/RC\in [-1:1]$ and Var$(\xi(t))\in[0.05:0.035]$ (delineated by the dashed rectangle in panel (a)). Other parameters are: $\gamma=0.2$, $\tau_1=1000$ (simulations) and $\tau_1=69.5$ ms (experiments). The additive force is absent, $F(t)\equiv 0$.
• Figure 5: Introducing the dual delayed feedback prevents wavefront propagation failure induced by additive noise, which is observed both in physical experiments (panels (a), (b)) and numerical simulations (panels (c), (d)): the spatial domain corresponding to $x(\sigma,n)=-b$ is destroyed in case $\tau_2=\tau_1$ (panels (a),(c)) but persists when the delayed feedback becomes dual ($\tau_2=0.99\tau_1$ in panel (b) and $\tau_2=0.945\tau_1$ in panel (d)). The upper insets show the instantaneous systems' state at discrete time moments $n=n_*$ in space-time plots: $n_*=550$ (panel (a)), $n_*=40$ (panel (b)), $n_*=90$ (panel (c)), $n_*=190$ (panel (d)). Parameters of model (\ref{['numerical_model']}) and setup (\ref{['experimental_model']}) are $a=0.5$, $b=0.45$, $\gamma=0.2$, $\tau_1=1000$ (simulations) and $\tau_1=69.5$ ms (experiments). The quasi-space length is $\eta=1005.95$ (simulations) and $\eta=69.8412$ ms (experiments). Additive noise is introduced as $F(t)=\xi(t)$ where $\xi(t)$ is a source of white Gaussian noise of variance Var$(\xi(t))=0.01$ (simulations) and Var$(\xi(t))=$ (experiments). The upper insets show the instantaneous systems' state at discrete time moments $n=n_*$ in space-time plots. The quasi-space length is $\eta=1005.95$ (simulations, panels (c),(d)) and $\eta=69.8412$ ms (experiments, panels (a),(b)).