arXiv subjects

Nils Berglund

Publications and source records attributed to Nils Berglund.

51 records · Page 3Linked to original sources

arXiv2007

Metastability in Interacting Nonlinear Stochastic Differential Equations I: From Weak Coupling to Synchronisation

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. We show that as the coupling strength increases, the number of equilibrium points of the system changes 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 oscillators 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, in which the metastable behaviour is encoded.

math.PR↗

arXiv2007

Metastability in Interacting Nonlinear Stochastic Differential Equations II: Large-N Behaviour

We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. 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. For strong coupling (of the order N^2), the system synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. In a previous paper, we showed that the transition from strong to weak coupling involves a sequence of symmetry-breaking bifurcations of the system's stationary configurations, and analysed in particular the behaviour for coupling intensities slightly below the synchronisation threshold, for arbitrary N. Here we describe the behaviour for any positive coupling intensity γof order N^2, provided the particle number N is sufficiently large (as a function of γ/N^2). In particular, we determine the transition time between synchronised states, as well as the shape of the "critical droplet", to leading order in 1/N. Our techniques involve the control of the exact number of periodic orbits of a near-integrable twist map, allowing us to give a detailed description of the system's potential landscape, in which the metastable behaviour is encoded.

math.PR↗

arXiv2005

Universality of residence-time distributions in non-adiabatic stochastic resonance

We present mathematically rigorous expressions for the residence-time and first-passage-time distributions of a periodically forced Brownian particle in a bistable potential. For a broad range of forcing frequencies and amplitudes, the distributions are close to periodically modulated exponential ones. Remarkably, the periodic modulations are governed by universal functions, depending on a single parameter related to the forcing period. The behaviour of the distributions and their moments is analysed, in particular in the low- and high-frequency limits.

cond-mat.dis-nn↗

arXiv2003

On the noise-induced passage through an unstable periodic orbit I: Two-level model

We consider the problem of stochastic exit from a planar domain, whose boundary is an unstable periodic orbit, and which contains a stable periodic orbit. This problem arises when investigating the distribution of noise-induced phase slips between synchronized oscillators, or when studying stochastic resonance far from the adiabatic limit. We introduce a simple, piecewise linear model equation, for which the distribution of first-passage times can be precisely computed. In particular, we obtain a quantitative description of the phenomenon of cycling: The distribution of first-passage times rotates around the unstable orbit, periodically in the logarithm of the noise intensity, and thus does not converge in the zero-noise limit. We compute explicitly the cycling profile, which is universal in the sense that in depends only on the product of the period of the unstable orbit with its Lyapunov exponent.

math.PR↗

arXiv2003

Geometric singular perturbation theory for stochastic differential equations

We consider slow-fast systems of differential equations, in which both the slow and fast variables are perturbed by noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold, which we construct explicitly. Depending on the dynamics of the reduced system, the results cover time spans which can be exponentially long in the noise intensity squared (that is, up to Kramers' time). We obtain exponentially small upper and lower bounds on the probability of exceptional paths. If the slow manifold contains bifurcation points, we show similar concentration properties for the fast variables corresponding to non-bifurcating modes. We also give conditions under which the system can be approximated by a lower-dimensional one, in which the fast variables contain only bifurcating modes.

math.PR↗

arXiv2001

Geometrical theory of dynamical systems

This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Winter term 2000-2001, to undergraduate Mathematics and Physics students. Contents: Chapter 1 - Examples of Dynamical Systems Chapter 2 - Stationary and Periodic Solutions Chapter 3 - Local Bifurcations Chapter 4 - Introduction to Chaotic Dynamics

math.HO↗

arXiv2001

Perturbation theory of dynamical systems

This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Summer term 2001, to undergraduate Mathematics and Physics students. It covers a few selected topics from perturbation theory at an introductory level. Only certain results are proved, and for some of the most important theorems, sketches of the proofs are provided. Contents: Chapter 1 - Introduction and Examples Chapter 2 - Bifurcations and Unfolding Chapter 3 - Regular Perturbation Theory Chapter 4 - Singular Perturbation Theory

math.HO↗

arXiv2001

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 behaviour 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 Ice Ages; hysteresis in a box model for the Atlantic thermohaline circulation; and bifurcation delay in the case of the Lorenz model for Rayleigh-B'enard convection.

physics.ao-ph↗

arXiv2001

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

We introduce a new method, allowing to describe slowly time-dependent Langevin equations through the behaviour of individual paths. This approach yields considerably more information than the computation of the probability density. The main idea is to show that for sufficiently small noise intensity and slow time dependence, the vast majority of paths remain in small space-time sets, typically in the neighbourhood of potential wells. The size of these sets often has a power-law dependence on the small parameters, with universal exponents. The overall probability of exceptional paths is exponentially small, with an exponent also showing power-law behaviour. The results cover time spans up to the maximal Kramers time of the system. We apply our method to three phenomena characteristic for bistable systems: stochastic resonance, dynamical hysteresis and bifurcation delay, where it yields precise bounds on transition probabilities, and the distribution of hysteresis areas and first-exit times. We also discuss the effect of coloured noise.

cond-mat.stat-mech↗

arXiv2001

The effect of additive noise on dynamical hysteresis

We investigate the properties of hysteresis cycles produced by a one-dimensional, periodically forced Langevin equation. We show that depending on amplitude and frequency of the forcing and on noise intensity, there are three qualitatively different types of hysteresis cycles. Below a critical noise intensity, the random area enclosed by hysteresis cycles is concentrated near the deterministic area, which is different for small and large driving amplitude. Above this threshold, the area of typical hysteresis cycles depends, to leading order, only on the noise intensity. In all three regimes, we derive mathematically rigorous estimates for expectation, variance, and the probability of deviations of the hysteresis area from its typical value.

math.DS↗

arXiv2000

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 $1/\eps$, where $\eps$ is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity $σ$ 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 $\eps$ and on the minimal barrier height, as well as the asymptotics of the transition and non-transition probabilities.

math.PR↗

arXiv2000

Pathwise description of dynamic pitchfork bifurcations with additive noise

The slow drift (with speed $\eps$) 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 $\sqrt\eps$ after the bifurcation, the paths are concentrated in a region of size $σ/\eps^{1/4}$ around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval $[\sqrt\eps, c\sqrt{\eps\abs{\logσ}}]$, 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.

math.PR↗

arXiv2000

The averaged dynamics of the hydrogen atom in crossed electric and magnetic fields as a perturbed Kepler problem

We treat the classical dynamics of the hydrogen atom in perpendicular electric and magnetic fields as a celestial mechanics problem. By expressing the Hamiltonian in appropriate action-angle variables, we separate the different time scales of the motion. The method of averaging then allows us to reduce the system to two degrees of freedom, and to classify the most important periodic orbits.

nlin.CD↗

arXiv1999

Control of Dynamic Hopf Bifurcations

The slow passage through a Hopf bifurcation leads to the delayed appearance of large amplitude oscillations. We construct a smooth scalar feedback control which suppresses the delay and causes the system to follow a stable equilibrium branch. This feature can be used to detect in time the loss of stability of an ageing device. As a by-product, we obtain results on the slow passage through a bifurcation with double zero eigenvalue, described by a singularly perturbed cubic Lienard equation.

chao-dyn↗

arXiv1998

Hysteresis in Adiabatic Dynamical Systems: an Introduction

We give a nontechnical description of the behaviour of dynamical systems governed by two distinct time scales. We discuss in particular memory effects, such as bifurcation delay and hysteresis, and comment the scaling behaviour of hysteresis cycles. These properties are illustrated on a few simple examples.

chao-dyn↗