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.

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.

We consider the Lorenz system near the classic parameter regime and study the phenomenon we call an α-flip. An α-flip is a transition where the onedimensional stable manifolds W s (p ± ) of two secondary equilibria p ± undergo a sudden transition in terms of the direction from which they approach p ± . This is a bifurcation at infinity and does not involve an invariant object in phase space. This fact was discovered by Sparrow in the 1980s but the stages of the transition could not be calculated and the phenomenon was not well understood (Sparrow 1982 The Lorenz equations (New York: Springer)). Here we employ a boundary value problem set-up and use pseudo-arclength continuation in AUTO to follow this sudden transition of W s (p ± ) as a continuous family of orbit segments. In this way, we geometrically characterize and determine the moment of the actual α-flip. We also investigate how the α-flip takes place relative to the two-dimensional stable manifold of the origin, which shows no apparent topological change before or after the α-flip. Our approach allows for easy detection and subsequent two-parameter continuation of the first and further α-flips. We illustrate this for the first 25 α-flips and find that they end at terminal points, or T-points, where there is a heteroclinic connection from the secondary equilibria to the origin. It turns out that α-flips must occur naturally near T-points. We find scaling relations for the α-flips and T-points that allow us to predict further such bifurcations and to improve the efficiency of our computations.

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