[Paper Review] Fast and slow domino effects in transient network dynamics

Peter Ashwin, Jennifer Creaser+1 more|arXiv (Cornell University)|Jan 22, 2017

Nonlinear Dynamics and Pattern Formation3 citations

TL;DR

This paper investigates noise-induced transitions in coupled multistable dynamical systems, identifying three distinct regimes—stochastic transitions, fast domino effects (synchronous transitions), and slow domino effects (delayed transitions)—based on coupling strength. In the low-noise limit, these regimes emerge from bifurcations in the potential landscape of the coupled system, revealing how network structure governs collective switching behavior.

ABSTRACT

It is well known that the addition of noise to a multistable dynamical system can induce random transitions from one stable state to another. For low noise, the times between transitions have an exponential tail and Kramers' escape time gives an expression for the mean escape time in the asymptotic limit. If a number of multistable systems are coupled into a network structure, a transition at one site may change the transition properties at other sites. We study the case of escape from a quiescent attractor to an active attractor in cases where transitions back can be ignored. We show that there are qualitatively different regimes of transition, depending on strength. For weak coupling the transition rates are simply modified but the transitions remain stochastic. For strong coupling transitions happen approximately in synchrony - we call this a fast domino effect, while in an intermediate coupling some transitions may happen inexorably but with some delay - we call this a slow domino effect. We characterise these regimes in the low noise limit in terms of bifurcations of the potential landscape of a specific coupled system.

Motivation & Objective

• To understand how coupling between multistable systems alters noise-induced transition dynamics in networks.
• To identify distinct dynamical regimes—stochastic, fast, and slow domino effects—based on coupling strength.
• To characterize these regimes using bifurcations in the potential landscape of the coupled system in the low-noise limit.
• To explain how local transitions propagate through a network, depending on coupling intensity.

Proposed method

• Model a network of coupled multistable systems where transitions from quiescent to active states are irreversible.
• Use the Fokker-Planck equation to describe the stochastic dynamics under low noise.
• Analyze the potential landscape of the coupled system to identify bifurcations that signal regime transitions.
• Apply asymptotic analysis to Kramers' escape time in the low-noise limit to derive mean transition times.
• Classify transition behavior into three regimes: weak coupling (modified stochastic rates), strong coupling (fast domino effect), intermediate coupling (slow domino effect).
• Use numerical and analytical tools to map the transition from stochastic to collective behavior via potential landscape changes.

Experimental results

Research questions

• RQ1 How does coupling strength influence the nature of noise-induced transitions in a network of multistable systems?
• RQ2 What distinguishes fast domino effects (synchronous transitions) from slow domino effects (delayed transitions) in coupled networks?
• RQ3 How do bifurcations in the potential landscape correspond to transitions between different dynamical regimes?
• RQ4 In what way does strong coupling lead to approximately synchronous transitions despite stochastic noise?
• RQ5 How does weak coupling modify transition rates without inducing collective behavior?

Key findings

• For weak coupling, transition rates are modified but transitions remain stochastic, with no significant synchronization.
• In strong coupling, transitions occur approximately in synchrony, leading to a fast domino effect due to mutual destabilization of stable states.
• In intermediate coupling, some transitions occur inexorably but with delays, resulting in a slow domino effect with sequential propagation.
• The three regimes—stochastic, slow, and fast domino effects—are characterized by distinct bifurcations in the potential landscape of the coupled system.
• The low-noise limit allows analytical classification of these regimes via bifurcation analysis of the effective potential.
• The transition from isolated to collective behavior is governed by the interplay between noise intensity and coupling strength, with the potential landscape serving as a key diagnostic tool.