Quasi symplectic integrators for stochastic differential equations

Two specialized algorithms for the numerical integration of the equations of motion of a Brownian walker obeying detailed balance are introduced. The algorithms become symplectic in the appropriate limits, and reproduce the equilibrium distributions to some higher order in the integration time step. Comparisons with other existing integration schemes are carried out both for static and dynamical quantities.Comment: 7 pages, revtex, 6 eps figure

Comment on "Influence of Noise on Force Measurements"

In a recent Letter [arXiv:1004.0874], Volpe et al. describe experiments on a colloidal particle near a wall in the presence of a gravitational field for which they study the influence of noise on the measurement of force. Their central result is a striking discrepancy between the forces derived from experimental drift measurements via their Eq. (1), and from the equilibrium distribution. From this discrepancy they infer the stochastic calculus realised in the system. We comment, however: (a) that Eq. (1) does not hold for space-dependent diffusion, and corrections should be introduced; and (b) that the "force" derived from the drift need not coincide with the "force" obtained from the equilibrium distribution.Comment: Comment submitted to a PRL letter; 1 page, 1 figur

Separatrix chaos: new approach to the theoretical treatment

We develop a new approach to the theoretical treatment of the separatrix chaos, using a special analysis of the separatrix map. The approach allows us to describe boundaries of the separatrix chaotic layer in the Poincar\'{e} section and transport within the layer. We show that the maximum which the width of the layer in energy takes as the perturbation frequency varies is much larger than the perturbation amplitude, in contrast to predictions by earlier theories suggesting that the maximum width is of the order of the amplitude. The approach has also allowed us to develop the self-consistent theory of the earlier discovered (PRL 90, 174101 (2003)) drastic facilitation of the onset of global chaos between adjacent separatrices. Simulations agree with the theory.Comment: 10 pages, 4 figures, proceedings of the conference "Chaos, Complexity and Transport" (Marseille, 5-9 June 2007), in pres

New approach to the treatment of separatrix chaos and its application to the global chaos onset between adjacent separatrices

We have developed the {\it general method} for the description of {\it separatrix chaos}, basing on the analysis of the separatrix map dynamics. Matching it with the resonant Hamiltonian analysis, we show that, for a given amplitude of perturbation, the maximum width of the chaotic layer in energy may be much larger than it was assumed before. We apply the above theory to explain the drastic facilitation of global chaos onset in time-periodically perturbed Hamiltonian systems possessing two or more separatrices, previously discovered (PRL 90, 174101 (2003)). The theory well agrees with simulations. We also discuss generalizations and applications. Examples of applications of the facilitation include: the increase of the DC conductivity in spatially periodic structures, the reduction of activation barriers for noise-induced transitions and the related acceleration of spatial diffusion, the facilitation of the stochastic web formation in a wave-driven or kicked oscillator.Comment: 29 pages, 16 figures (figs. are of reduced quality, original files are available on request from authors), paper has been significantly revised and resubmitted to PR

On the determination of the optimal parameters in the CAM model

In the field of complex systems, it is often possible to arrive at some simple stochastic or chaotic Low Order Models (LOMs) exploiting the time scale separation between leading modes of interest and fast fluctuations. These LOMs, although approximate, might provide interesting qualitative insights regarding some important aspects like the average time between two extreme events. Recently, the simplest example of a LOM with multiplicative noise, namely, a linear system with a linearly state dependent noise [also called correlated additive and multiplicative (CAM) model], has been considered as archetypal for numerous phenomena that present markedly non-Gaussian statistics. We show in this paper that the determination of the parameters of a CAM model from the (few) available data is far from trivial and that the actual most likely parameters might differ substantially from the ones determined directly from a (necessarily limited) short sequence of observations. We illustrate how this problem can be tackled, at least to the extent possible, using an approach that is based on Bayes' theorem. We shall focus on a CAM modeling the El Ninõ Southern Oscillation but the methodology can be extended to any phenomenon that can be described by a simplified LOM similar to the one examined here and where the available sequence of data is relatively short. We conclude that indeed a Bayesian approach can fix the problem

Fast Monte Carlo simulations and singularities in the probability distributions of non-equilibrium systems

A numerical technique is introduced that reduces exponentially the time required for Monte Carlo simulations of non-equilibrium systems. Results for the quasi-stationary probability distribution in two model systems are compared with the asymptotically exact theory in the limit of extremely small noise intensity. Singularities of the non-equilibrium distributions are revealed by the simulations.Comment: 4 pages, 4 figure

Stochastic resonance in electrical circuits—II: Nonconventional stochastic resonance.

Stochastic resonance (SR), in which a periodic signal in a nonlinear system can be amplified by added noise, is discussed. The application of circuit modeling techniques to the conventional form of SR, which occurs in static bistable potentials, was considered in a companion paper. Here, the investigation of nonconventional forms of SR in part using similar electronic techniques is described. In the small-signal limit, the results are well described in terms of linear response theory. Some other phenomena of topical interest, closely related to SR, are also treate

Enlargement of a low-dimensional stochastic web

We consider an archetypal example of a low-dimensional stochastic web, arising in a 1D oscillator driven by a plane wave of a frequency equal or close to a multiple of the oscillator’s natural frequency. We show that the web can be greatly enlarged by the introduction of a slow, very weak, modulation of the wave angle. Generalizations are discussed. An application to electron transport in a nanometre-scale semiconductor superlattice in electric and magnetic fields is suggested

Stochastic resonance in electrical circuits—I: Conventional stochastic resonance.

Stochastic resonance (SR), a phenomenon in which a periodic signal in a nonlinear system can be amplified by added noise, is introduced and discussed. Techniques for investigating SR using electronic circuits are described in practical terms. The physical nature of SR, and the explanation of weak-noise SR as a linear response phenomenon, are considered. Conventional SR, for systems characterized by static bistable potentials, is described together with examples of the data obtainable from the circuit models used to test the theory