Chaotic stochastic resonance in Mackey-Glass equations

TL;DR

The paper investigates noise-induced resonance in the stochastic Mackey–Glass delay differential equation using random dynamical systems theory, distinguishing stable SR (random point attractors with $\lambda_{\max}<0$) from chaotic SR (random strange attractors with $\lambda_{\max}>0$). It demonstrates both SR types in MG, and shows chaotic SR also occurs in the Duffing and underdamped FitzHugh–Nagumo equations, suggesting universality across strongly nonlinear random systems with delay. By analyzing random Lyapunov spectra and random pullback attractors, the authors characterize resonance through the relation between the resonance noise level $\sigma^*$ and the LE zero-crossing $\sigma_0$, and refine the resonant period as $T \approx \tau(1+\varepsilon)$ with $\varepsilon = 1/[b\tau + \ln(a\tau) - 1]$, linking SR to noise-driven visits of unstable spirals. The work provides a framework for understanding how stochastic chaos and resonance interact in delay systems, with potential implications for predicting and controlling noise-enhanced dynamics in complex biological and physical networks.

Abstract

Stochastic resonance (SR) manifests as switching dynamics between two quasi-stationary states in the stochastic Mackey-Glass equation. We identify chaotic SR, arising from the coexistence of resonance and chaos in stochastic dynamics. In contrast to classical SR, which is described by a random point attractor with a negative largest Lyapunov exponent, chaotic SR is described by a random strange attractor with a positive largest Lyapunov exponent. We observe chaotic SR in the Mackey-Glass equation as well as chaotic SR in the Duffing equation and the underdamped FitzHugh-Nagumo equation, demonstrating the universality of this phenomenon across a broad class of strongly nonlinear random dynamical systems.

Figures (9)

• Figure 1: (a) Bifurcation diagram of the MG with $\sigma = 0$. (b) Bifurcation diagram of random pullback attractors of the SMG with $\sigma = 0.15$. (c) First and second largest LEs for $\sigma = 0$ (blue points) and $\sigma = 0.15$ (black points). In the bifurcation diagram, random pullback attractors are computed using $1.0\times 10^2$ constant initial functions uniformly distributed in $[-1,1]$ with a pullback time $t_p = 4\times10^4$. We designate $a_1=0.14$ as a representative example of a weakly nonlinear regime and $a_2=0.30$ as that of a strongly nonlinear regime. At $a=a_1, a_2$, low- and high-dimensional deterministic chaos are observed, respectively. The values, $a_c$ and $a_d$, indicate the zero-crossing points of the first and second largest LEs in the presence of noise. The LEs and the bifurcation diagram were computed using a time step of $\Delta t=0.1$ and a resolution defined as $N := \tau/\Delta t = 900$. Note that for delay differential equations, the largest LEs are known to be well approximated with a limited resolution of $N= \tau/\Delta t\sim 100$wernecke2019chaos.
• Figure 2: (a) Two deterministic attractors projected onto $(x(t),x(t-\tau))$ plane for $a=0.126$. The inset shows a magnified view of the limit cycle. (b) Time series for the deterministic case, $\sigma=0$ (blue dots), and the resonance case, $\sigma = \sigma^*$ (black dots). (c) Power spectra for $\sigma=0$ (blue lines) and the optimal noise intensity $\sigma=\sigma^*$ (black lines). (d) Power at the primary resonant frequency $f_1^*$ as a function of $\sigma$.
• Figure 3: Power spectra for $\sigma=0$ (blue lines) and $\sigma=\sigma^*$ (black lines) at (a) $a=a_1$ and (c) $a=a_2$. Power at the primary resonant frequency $f_1^*$ (black points) as a function of $\sigma$ at (b) $a=a_1$ and (d) $a=a_2$. The largest random LE at $a=a_1$ and the largest six random LEs at $a=a_2$ are also shown in (b) and (d), respectively (red points). The random LEs were computed using a time step of $\Delta t=0.1$ and a resolution defined as $N := \tau/\Delta t = 900$.
• Figure 4: Space--time representation at the resonance point for (a) stable SR ($a = a_1, \sigma = 0.162$) and (c) chaotic SR ($a = a_2, \sigma = 0.104$). In both cases, traveling waves appear in the memory space $s\in[0,\tau)$. The angle $\theta$ denotes the propagation direction of the traveling wave relative to the orthogonal direction, defined by $\tan\theta=\epsilon$, where $\epsilon$ represents the deviation of the resonant period (see Eq. (\ref{['eq:frequency deviation']})). From the measured angles $\theta_1$ (stable SR) and $\theta_2$ (chaotic SR), we obtain $\theta_2/\theta_1 \simeq 0.786$, indicating that wave propagation is slower in the chaotic SR regime. Panels (b) and (d) show snapshots of the random pullback attractor at the resonance point computed from $1.0\times 10^5$ constant initial functions uniformly distributed on $[-1,1]$ with a pullback time $t_p = 2\times10^4$, projected onto $(x(t), x(t-\tau))$ plane.
• Figure 5: Unstable spirals in the weakly and strongly nonlinear regimes. Phase-space portraits projected onto $(x(t), x(t-\tau))$ plane for (a) $a = a_1$ and (c) $a = a_2$. Transients initialized near the origin (red dots) eventually converge to other stable attractors (blue dots). The trajectory is computed from the initial function $\phi(t)= 0.01 \sin(2\pi f^*_1 t) + 0.02 \sin(2\pi f^*_2 t)$ and departs from the origin following a spiral structure. The primary resonant frequency $f_1^*$ as a function of $\tau$ for (b) $a = a_1$ and (d) $a = a_2$. The theoretical estimate, the reciprocal of Eq. (\ref{['eq:frequency deviation']}), (red lines) agrees well with numerical results (black points). The classical estimate $f_1^* = 1/\tau$ is also shown (dotted lines).