Qualitatively distinct mechanisms of noise-induced escape in diffusively coupled bistable elements

TL;DR

This paper tackles noise-induced escape in ensembles of diffusively coupled bistable elements by showing that coupling strength $K$ gives rise to three qualitatively distinct escape mechanisms. The authors develop three reduced one-dimensional dynamics—NlinMFFPE for weak coupling, DMFD for intermediate coupling, and SMFD for strong coupling—and verify them against direct simulations, revealing regime-specific dominant escape drivers. A key insight is that these mechanisms arise from the interplay of nonlinearity, diffusion, and dynamical noise rather than noise-free bifurcations, with regime boundaries $K_1$ and $K_2$ delimiting the transitions. The work provides a scalable framework applicable to other diffusively coupled stochastic nonlinear systems and networks, enabling broader exploration of synergistic phenomena in collective escape.

Abstract

The analysis of noise-induced escape in ensembles of bistable elements is challenging, because nonlinearity, coupling, and noise all play essential roles. We show that the interplay of these three factors yields three qualitatively distinct escape mechanisms in diffusively coupled bistable elements, depending on the coupling strength. To clarify the relation between coupling strength and mean escape time, we derive effective one-dimensional dynamics: nonlinear mean-field Fokker-Planck equation in the weak-coupling regime, stochastic mean-field dynamics in the strong-coupling regime, and deterministic mean-field dynamics in the intermediate regime. We validate these reduced descriptions by comparing predictions with numerical simulations. We identify a distinct dominant driving factor of escape processes in each regime. Notably, the three escape mechanisms emerge through the interplay of nonlinearity, diffusive coupling, and dynamical noise -- rather than bifurcations of the noise-free system. Our approach serves as a framework applicable to other diffusively coupled stochastic nonlinear systems, motivating a further search for similar synergistic phenomena.

Figures (4)

• Figure 1: Typical trajectories in different regimes. Light gray lines depict the evolution of all the elements. Bright red lines show trajectories of the mean field. (a) Elements are non-synchronous in the weak-coupling regime. (b) In the intermediate regime, elements' states evolve around the mean field while maintaining non-negligible variance. (c) The ensemble behaves as a single unit under strong diffusive coupling.
• Figure 2: Bifurcation diagram of deterministic mean-field dynamics (DMFD). The system is monostable for $K < K_2 \approx 7.99$ and bistable otherwise.
• Figure 3: The effective potential $V_\mathrm{1d}(x; X, K)$ for $X = 0$, with different values of $K$. While $V_\mathrm{1d}$ has two minima when $K$ is small, the influence of diffusive coupling becomes more pronounced for larger $K$, eventually making the system monostable.
• Figure 4: Numerically measured mean escape times (markers) and theoretical predictions (lines) against coupling strength $K$. Numerical results agreed well with our theoretical predictions. Background colors distinguish the three regimes (weak-coupling: green, intermediate: purple, and strong-coupling: yellow), separated by $K_1$ and $K_2$. The horizontal dashed line $T_0$ indicates the mean escape time of the uncoupled system ($K = 0$). (a) Weak-coupling and intermediate regimes. (b) Intermediate and strong-coupling regimes. The case of $N = 1024$ was omitted.