Nils Berglund

METASTABILITY IN SIMPLE CLIMATE MODELS: PATHWISE ANALYSIS OF SLOWLY DRIVEN LANGEVIN EQUATIONS

We consider simple stochastic climate models, described by slowly time-dependent Langevin equations. We show that when the noise intensity is not too large, these systems can spend substantial amounts of time in metastable equilibrium, instead of adiabatically following the stationary distribution of the frozen system. This behavior can be characterized by describing the location of typical paths, and bounding the probability of atypical paths. We illustrate this approach by giving a quantitative description of phenomena associated with bistability, for three famous examples of simple climate models: Stochastic resonance in an energy balance model describing the Ice Ages; hysteresis in a box model for the Atlantic thermohaline circulation; and bifurcation delay in the case of the Lorenz model for Rayleigh–Benard convection.

Pathwise description of dynamic pitchfork bifurcations with additive noise

Abstract. The slow drift (with speed ɛ) of a parameter through a pitchfork bifurcation point, known as the dynamic pitchfork bifurcation, is characterized by a significant delay of the transition from the unstable to the stable state. We describe the effect of an additive noise, of intensity σ, by giving precise estimates on the behaviour of the individual paths. We show that until time after the bifurcation, the paths are concentrated in a region of size around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval , after which they are likely to stay close to the corresponding deterministic solution. We derive exponentially small upper bounds for the probability of the sets of exceptional paths, with explicit values for the exponents.

Integrability and Ergodicity of Classical Billiards in a Magnetic Field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a “bouncing map”. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.

A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential

Additive white noise may significantly increase the response of bistable systems to a periodic driving signal. We consider two classes of double-well potentials, symmetric and asymmetric, modulated periodically in time with period

Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh–Nagumo model

1/\eps

On the Noise-Induced Passage Through an Unstable Periodic Orbit I: Two-Level Model

, where

Memory effects and scaling laws in slowly driven systems

\eps

Hunting French Ducks in a Noisy Environment

is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity

Beyond the Fokker-Planck equation: pathwise control of noisy bistable systems

\sigma

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

crosses a threshold value. Below the threshold, paths are concentrated near one potential well, and have an exponentially small probability to jump to the other well. Above the threshold, transitions between the wells occur with probability exponentially close to 1/2 in the symmetric case, and exponentially close to 1 in the asymmetric case. The transition zones are localised in time near the points of minimal barrier height. We give a mathematically rigorous description of the behaviour of individual paths, which allows us, in particular, to determine the power-law dependence of the critical noise intensity on