Inference of stochastic nonlinear oscillators with applications to physiological problems

A new method of inferencing of coupled stochastic nonlinear oscillators is described. The technique does not require extensive global optimization, provides optimal compensation for noise-induced errors and is robust in a broad range of dynamical models. We illustrate the main ideas of the technique by inferencing a model of five globally and locally coupled noisy oscillators. Specific modifications of the technique for inferencing hidden degrees of freedom of coupled nonlinear oscillators is discussed in the context of physiological applications.Comment: 11 pages, 10 figures, 2 tables Fluctuations and Noise 2004, SPIE Conference, 25-28 May 2004 Gran Hotel Costa Meloneras Maspalomas, Gran Canaria, Spai

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

We develop a moment equation closure minimization method for the inexpensive approximation of the steady state statistical structure of nonlinear systems whose potential functions have bimodal shapes and which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics that has i) single time statistical structure consistent with the analytical solutions of the Fokker-Planck equation, and ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-of-freedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo Simulations

Low noise all-fiber amplification of a coherent supercontinuum at 2 \mu m and its limits imposed by polarization noise

We report the amplification of an all-normal dispersion supercontinuum pulse in a Thulium / Holmium co-doped all-fiber chirped pulse amplification system. With a -20 dB bandwidth of more than 300 nm in the range 1800-2100 nm the system delivers high quality 66 fs pulses with more than 70 kW peak power directly from the output fiber. The coherent seeding of the entire emission bandwidth of the doped fiber and the stability of the supercontinuum generation dynamics in the silicate glass all-normal dispersion photonic crystal fiber result in excellent noise characteristics of the amplified ultrashort pulses

Energy Harvesting From Bistable Systems Under Random Excitation

The transformation of otherwise unused vibrational energy into electric energy through the use of piezoelectric energy harvesting devices has been the subject of numerous investigations. The mechanical part of such a device is often constructed as a cantilever beam with applied piezo patches. If the harvester is designed as a linear resonator the power output relies strongly on the matching of the natural frequency of the beam and the frequency of the harvested vibration which restricts the applicability since most vibrations which are found in built environments are broad-banded or stochastic in nature. A possible approach to overcome this restriction is the use of permanent magnets to impose a nonlinear restoring force on the beam that leads to a broader operating range due to large amplitude motions over a large range of excitation frequencies. In this paper such a system is considered introducing a refined modeling with a modal expansion that incorporates two modal functions and a refined modeling of the magnet beam interaction. The corresponding probability density function in case of random excitation is calculated by the solution of the corresponding Fokker-Planck equation and compared with results from Monte Carlo simulations. Finally some measurements of ambient excitations are discussed.DFG, 253161314, Untersuchung des nichtlinearen dynamischen Verhaltens von stochastisch erregten Energy Harvesting Systemen mittels Lösung der Fokker-Planck-Gleichun

Some dynamics of acoustic oscillations with nonlinear combustion and noise

The results given in this paper constitute a continuation of progress with nonlinear analysis of coherent oscillations in combustion chambers. We are currently focusing attention on two general problems of nonlinear behavior important to practical applications: the conditions under which a linearly unstable system will execute stable periodic limit cycles; and the conditions under which a linearly stable system is unstable to a sufficiently large disturbance. The first of these is often called 'soft' excitation, or supercritical bifurcation; the second is called 'hard' excitation, 'triggering,' or subcritical bifurcation and is the focus of this paper. Previous works extending over more than a decade have established beyond serious doubt (although no formal proof exists) that nonlinear gasdynamics alone does not contain subcritical bifurcations. The present work has shown that nonlinear combustion alone also does not contain subcritical bifurcations, but the combination of nonlinear gasdynamics and combustion does. Some examples are given for simple models of nonlinear combustion of a solid propellant but the broad conclusion just mentioned is valid for any combustion system. Although flows in combustors contain considerable noise, arising from several kinds of sources, there is sound basis for treating organized oscillations as distinct motions. That has been an essential assumption incorporated in virtually all treatments of combustion instabilities. However, certain characteristics of the organized or deterministic motions seem to have the nature of stochastic processes. For example, the amplitudes in limit cycles always exhibit a random character and even the occurrence of instabilities seems occasionally to possess some statistical features. Analysis of nonlinear coherent motions in the presence of stochastic sources is therefore an important part of the theory. We report here a few results of power spectral densities of acoustic amplitudes in the presence of a subcritical bifurcation associated with nonlinear combustion and gasdynamics

A Passive Phase Noise Cancellation Element

We introduce a new method for reducing phase noise in oscillators, thereby improving their frequency precision. The noise reduction device consists of a pair of coupled nonlinear resonating elements that are driven parametrically by the output of a conventional oscillator at a frequency close to the sum of the linear mode frequencies. Above the threshold for parametric response, the coupled resonators exhibit self-oscillation at an inherent frequency. We find operating points of the device for which this periodic signal is immune to frequency noise in the driving oscillator, providing a way to clean its phase noise. We present results for the effect of thermal noise to advance a broader understanding of the overall noise sensitivity and the fundamental operating limits

One-Dimensional Population Density Approaches to Recurrently Coupled Networks of Neurons with Noise

Mean-field systems have been previously derived for networks of coupled, two-dimensional, integrate-and-fire neurons such as the Izhikevich, adapting exponential (AdEx) and quartic integrate and fire (QIF), among others. Unfortunately, the mean-field systems have a degree of frequency error and the networks analyzed often do not include noise when there is adaptation. Here, we derive a one-dimensional partial differential equation (PDE) approximation for the marginal voltage density under a first order moment closure for coupled networks of integrate-and-fire neurons with white noise inputs. The PDE has substantially less frequency error than the mean-field system, and provides a great deal more information, at the cost of analytical tractability. The convergence properties of the mean-field system in the low noise limit are elucidated. A novel method for the analysis of the stability of the asynchronous tonic firing solution is also presented and implemented. Unlike previous attempts at stability analysis with these network types, information about the marginal densities of the adaptation variables is used. This method can in principle be applied to other systems with nonlinear partial differential equations.Comment: 26 Pages, 6 Figure

Phase transitions driven by L\'evy stable noise: exact solutions and stability analysis of nonlinear fractional Fokker-Planck equations

Phase transitions and effects of external noise on many body systems are one of the main topics in physics. In mean field coupled nonlinear dynamical stochastic systems driven by Brownian noise, various types of phase transitions including nonequilibrium ones may appear. A Brownian motion is a special case of L\'evy motion and the stochastic process based on the latter is an alternative choice for studying cooperative phenomena in various fields. Recently, fractional Fokker-Planck equations associated with L\'evy noise have attracted much attention and behaviors of systems with double-well potential subjected to L\'evy noise have been studied intensively. However, most of such studies have resorted to numerical computation. We construct an {\it analytically solvable model} to study the occurrence of phase transitions driven by L\'evy stable noise.Comment: submitted to EP