Metastability in interacting nonlinear stochastic differential equations: II. Large-<i>N</i>behaviour

Berglund, Nils; Fernandez, Bastien; Gentz, Barbara

doi:10.1088/0951-7715/20/11/007

Cited by 26 publications

(41 citation statements)

References 19 publications

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“…Numerically, we checked the validity of the conjecture for N up to 101. In [BFG06b], we show that the conjecture is also true for sufficiently large N .…”

Section: )mentioning

confidence: 64%

Metastability in interacting nonlinear stochastic differential equations: I. From weak coupling to synchronization

Berglund¹,

Fernandez²,

Gentz³

2007

Nonlinearity

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We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. The system shows a metastable behaviour, which is characterised by the location and stability of its equilibrium points. We show that as the coupling strength increases, the number of equilibrium points decreases from 3 N to 3. While for weak coupling, the system behaves like an Ising model with spin-flip dynamics, for strong coupling (of the order N 2 ), it synchronises, in the sense that all particles assume almost the same position in their respective local potential most of the time. We derive the exponential asymptotics for the transition times, and describe the most probable transition paths between synchronised states, in particular for coupling intensities below the synchronisation threshold. Our techniques involve a centre-manifold analysis of the desynchronisation bifurcation, with a precise control of the stability of bifurcating solutions, allowing us to give a detailed description of the system's potential landscape.

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“…Numerically, we checked the validity of the conjecture for N up to 101. In [BFG06b], we show that the conjecture is also true for sufficiently large N .…”

Section: )mentioning

confidence: 64%

Metastability in interacting nonlinear stochastic differential equations: I. From weak coupling to synchronization

Berglund¹,

Fernandez²,

Gentz³

2007

Nonlinearity

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“…The role of coupling strength in noise-induced transitions on networks is considered by [6,7] for idealised symmetric bistable systems. Neiman [31] shows similar synchronization effects in coupled stochastic bistable systems and [30] in coupled ratchet systems.…”

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confidence: 99%

Fast and slow domino regimes in transient network dynamics

2017

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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' formula 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 which transitions back can be ignored. There are qualitatively different regimes of transition, depending on coupling strength. For small coupling strengths, the transition rates are simply modified but the transitions remain stochastic. For large coupling strengths, transitions happen approximately in synchrony-we call this a "fast domino" regime. There is also an intermediate coupling regime where some transitions happen inexorably but with a delay that may be arbitrarily long-we call this a "slow domino" regime. We characterize these regimes in the low noise limit in terms of bifurcations of the potential landscape of a coupled system. We demonstrate the effect of the coupling on the distribution of timings and (in general) the sequences of escapes of the system.

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“…In particular for strong bidirectional coupling we find (somewhat counterintuitively) that the mean escape time for one node is greatly increased by the coupling, but the mean escape time of the second node is greatly reduced: this completes the work presented in [6] that concerns only the first escape of one of the nodes. As previously discussed in a symmetric context [8,9] this behavior is due to the presence of bifurcations in the basin boundary of the steady attractor that correspond to synchronisation, though here the phase dynamics of the coupled oscillations adds an extra complication.…”

Section: Introductionmentioning

confidence: 78%

“…Coupling of such systems can promote (or decrease) escape of others on the network. However, there may also be critical values of the coupling, as identified in [3,8,9], at which the qualitative nature of the escape changes associated with bifurcations on basin boundaries of the attractors where typical transitions occur. As coupling strength increases, the escape of a node is increasingly dependent on the input from other nodes in addition to the noise perturbations.…”

Section: Introductionmentioning

confidence: 99%

Sequential Noise-Induced Escapes for Oscillatory Network Dynamics

Creaser¹,

Tsaneva-Atanasova²,

Ashwin³

2018

SIAM J. Appl. Dyn. Syst.

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It is well known that the addition of noise in a multistable system can induce random transitions between stable states. The rate of transition can be characterised in terms of the noise-free system's dynamics and the added noise: for potential systems in the presence of asymptotically low noise the well-known Kramers' escape time gives an expression for the mean escape time. This paper examines some general properties and examples of transitions between local steady and oscillatory attractors within networks: the transition rates at each node may be affected by the dynamics at other nodes. We use first passage time theory to explain some properties of scalings noted in the literature for an idealised model of initiation of epileptic seizures in small systems of coupled bistable systems with both steady and oscillatory attractors. We focus on the case of sequential escapes where a steady attractor is only marginally stable but all nodes start in this state. As the nodes escape to the oscillatory regime, we assume that the transitions back are very infrequent in comparison. We quantify and characterise the resulting sequences of noise-induced escapes. For weak enough coupling we show that a master equation approach gives a good quantitative understanding of sequential escapes, but for strong coupling this description breaks down.

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Metastability in interacting nonlinear stochastic differential equations: II. Large-Nbehaviour

Cited by 26 publications

References 19 publications

Metastability in interacting nonlinear stochastic differential equations: I. From weak coupling to synchronization

Metastability in interacting nonlinear stochastic differential equations: I. From weak coupling to synchronization

Fast and slow domino regimes in transient network dynamics

Sequential Noise-Induced Escapes for Oscillatory Network Dynamics

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