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.Comment: 30 pages, 5 figure

On the noise-induced passage through an unstable periodic orbit II: General case

Consider a dynamical system given by a planar differential equation, which exhibits an unstable periodic orbit surrounding a stable periodic orbit. It is known that under random perturbations, the distribution of locations where the system's first exit from the interior of the unstable orbit occurs, typically displays the phenomenon of cycling: The distribution of first-exit locations is translated along the unstable periodic orbit proportionally to the logarithm of the noise intensity as the noise intensity goes to zero. We show that for a large class of such systems, the cycling profile is given, up to a model-dependent change of coordinates, by a universal function given by a periodicised Gumbel distribution. Our techniques combine action-functional or large-deviation results with properties of random Poincar\'e maps described by continuous-space discrete-time Markov chains.Comment: 44 pages, 4 figure

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.Comment: 8 pages, 1 figure New version includes distinction between first-passage-time and residence-time distribution

Lemon Factory Extension

Martin works for Urban Future Organization (UFO), an internationally networked architectural practice involved in advanced digital design and fabrication. This new administrative wing is being added to an existing factory outside Messina in Sicily, sitting adjacent to a production building which is actually Italy’s largest producer of lemon juice and lemon extracts / essences. The brief was for a sequence of flexible spaces to house the reception area, staff offices, meeting spaces and canteen, along with a swimming pool and a fitness centre. The interior of the building reacts to the requirement to keep the spaces fluid and able to change their use over time. The project is currently on site and scheduled for completion in Spring 2008. In terms of research questions investigated, the key ones were how to develop new techniques of modulation design and structural design in what is a highly active earthquake zone, and then – given this crucial demand – how to create a new kind of flexible spatial organisation for a rapidly evolving company. In its design processes, the Lemon Factory has to be understood as part of a line of projects being carried out by the UFO practice and by similar entities – such as Ocean or Foreign Office Architects – into free-form, fluid and linear architectural forms, allying to this the pursuit of new forms of digital design and manufacturing in architecture. This project has been exhibited like other UFO project in important events such as the 2004 Venice Biennale, 2006 Beijing Biennale, and also in books like the press through articles such as in Building Design (7 March 2003, pp. 12-15). Urban Future Organization is a collaborative practice in which Martin and Yau are senior design figures, both being equally responsible for designing major projects such as the Lemon Factory near Messina

Modulated amplitude waves with nonzero phases in Bose-Einstein condensates

In this paper we give a frame for application of the averaging method to Bose-Einstein condensates (BECs) and obtain an abstract result upon the dynamics of BECs. Using aver- aging method, we determine the location where the modulated amplitude waves (periodic or quasi-periodic) exist and we also study the stability and instability of modulated amplitude waves (periodic or quasi-periodic). Compared with the previous work, modulated amplitude waves studied in this paper have nontrivial phases and this makes the problem become more diffcult, since it involves some singularities.Comment: 17 pages, 2 figure

Memory Effects and Scaling Laws in Slowly Driven Systems

This article deals with dynamical systems depending on a slowly varying parameter. We present several physical examples illustrating memory effects, such as metastability and hysteresis, which frequently appear in these systems. A mathematical theory is outlined, which allows to show existence of hysteresis cycles, and determine related scaling laws.Comment: 28 pages (AMS-LaTeX), 18 PS figure

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 \gamma of order N^2, provided the particle number N is sufficiently large (as a function of \gamma/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

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.Comment: 37 pages, 11 figure