Effect of noise and perturbations on limit cycle systems

This paper demonstrates that the influence of noise and of external perturbations on a nonlinear oscillator can vary strongly along the limit cycle and upon transition from limit cycle to stationary point behaviour. For this purpose we consider the role of noise on the Bonhoeffer-van der Pol model in a range of control parameters where the model exhibits a limit cycle, but the parameters are close to values corresponding to a stable stationary point. Our analysis is based on the van Kampen approximation for solutions of the Fokker-Planck equation in the limit of weak noise. We investigate first separately the effect of noise on motion tangential and normal to the limit cycle. The key result is that noise induces diffusion-like spread along the limit cycle, but quasistationary behaviour normal to the limit cycle. We then describe the coupled motion and show that noise acting in the normal direction can strongly enhance diffusion along the limit cycle. We finally argue that the variability of the system's response to noise can be exploited in populations of nonlinear oscillators in that weak coupling can induce synchronization as long as the single oscillators operate in a regime close to the transition between oscillatory and excitatory modes.     

A simple and effective Verlet-type algorithm for simulating Langevin dynamics

We present a revision to the well known Störmer–Verlet algorithm for simulating second order differential equations. The revision addresses the inclusion of linear friction with associated stochastic noise, and we analytically demonstrate that the new algorithm correctly reproduces diffusive behaviour of a particle in a flat potential. For a harmonic oscillator, our algorithm provides the exact Boltzmann distribution for any value of damping, frequency and time step for both underdamped and overdamped behaviour within the usual stability limit of the Verlet algorithm. Given the structure and simplicity of the method, we conclude that this approach can trivially be adapted for contemporary applications, including molecular dynamics with extensions such as molecular constraints.     

Coloured noise controlled dynamics of nonlinear polaritons in semiconductor microcavity

The coloured noise induced escape rate from the lower energy stable state of a driven nonlinear microcavity oscillator has been investigated by means of quasi-classical kinetic equations. We show that for coloured, i.e. narrow-band, relatively intense noise, the escape time is controlled by the interplay of two mechanisms: the noise induced drift and adiabatic regular shift of the oscillator state towards unstable saddle point. The cross-over between these mechanisms takes place in a particular range of the driving field intensity values, depending on the ratio between the oscillator damping and the coloured noise spectrum width. The dependence of the transition rate on the noise correlation time is analyzed for wide range of correlation time values. The article is published in the original.     

Asymmetry of Coordinate and Velocity in Noisy Oscillator

The velocity variable of a noise oscillator as an internal or external noise is proposed, the spectrum of which is quite different from that of the coordinate variable of the same noise oscillator. The former leads to ballistic diffusion for a free particle in long time limit and the asymptotical results of the system are sensitive to the initial condition. However, diffusion process induced by the coordinate of noise oscillator is a normal one and depends on the initial condition only in the transient time. This allows us to classify two kinds of non-Markovian processes: normal one and strong one, just like the processes induced by the coordinate and the velocity of noise oscillator, respectively. Applying to a correlation ratchet, we have found that the steady current of a particle subjected to the velocity of noise oscillator is opposite to that subjected to its coordinate, thus the former shows greenness and the latter redness.     

Hypersensitivity to small signals in a stochastic system with multiplicative colored noise

Recently we discovered the phenomenon of hypersensitivity to small time-dependent signals in a simple stochastic system, the Kramers oscillator with multiplicative white noise. In the present work we study, theoretically and experimentally with analog simulations, an influence of noise correlation time on hypersensitivity in a nonlinear oscillator with piecewise-linear current-voltage characteristic and multiplicative colored dichotomous noise. We found that the region of hypersensitive behavior is defined by universal scaling index, whereas the specifics of a particular system reveals itself only in the dependence of the above index on system parameters. The dependence of gain factor on noise correlation time is of bell-shaped (resonant) type. Received 27 April 2000 and Received in final form 2 November 2000     

Scaling behavior of a nonlinear oscillator with additive noise,white and colored

We study analytically and numerically the problem of a nonlinear mechanical oscillator with additive noise in the absence of damping. We show that the amplitude, the velocity and the energy of the oscillator grow algebraically with time. For Gaussian white noise, an analytical expression for the probability distribution function of the energy is obtained in the long-time limit. In the case of colored, Ornstein-Uhlenbeck noise, a self-consistent calculation leads to (different) anomalous diffusion exponents. Dimensional analysis yields the qualitative behavior of the prefactors (generalized diffusion constants) as a function of the correlation time. Received 10 October 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: mallick@spht.saclay.cea.fr     

Features of the moment functions of an oscillator with parametric instability due to dichotomous noise with Erlang distribution functions

We study stability of the periodic solutions of a linear undamped oscillator with frequency modulated by dichotomous noise whose statistic is determined by the Erlang distribution. It is shown that the amplitudes of harmonic oscillations of such an oscillator increase with time at different rates determined by the ratio of the oscillator eigenfrequency to the characteristic frequency of dichotomous noise. To solve the problem, we use a finite system of closed equations with respect to the moment functions, which was obtained without assumption on the quasi-Gaussianity and delta-correlatedness of the studied process.     

Phase reduction of stochastic limit cycle oscillators

We point out that for an oscillator subjected to noise the conventional phase equation is not a proper approximation even for weak noise. We present a phase reduction method valid for an oscillator subjected to weak white Gaussian noise. Numerical evidence demonstrates that the phase equation properly approximates dynamics of the original oscillator. Moreover, we show that, in general, noise causes a shift of the oscillator frequency and discuss its effects on entrainment.     

New Type of Brownian Motion

We consider an oscillator with a random mass for which the particles of the surrounding medium adhere to the oscillator for some random time after the collision (Brownian motion with adhesion for a harmonically bound particle). This is another form of a stochastic oscillator, different from oscillator usually studied that is subject to a random force or having random frequency or random damping. Calculation of the first two stationary moments shows that for white multiplicative noise of week strength the second moment coincides with that of usual Brownian motion, but for symmetric dichotomous noise, the second moment may appear the same type of the “energetic” instability, which exists for white noise random frequency or damping coefficient.     

Noise Effects on Oscillator Network of Transcription Regulators

Based on the model describing the regulation of the PRM operator region of λ phage proposed by Jeff Hastyet al., we study the noise effects on the oscillator network. We find that the additive noise cannot change the period andthe amplitude of the relaxation oscillator, but in the multiplicative case, the period of the relaxation oscillator increasesto a constant value with the increase of the strength of noise, and the amplitude of the relaxation oscillator also showsincreases with the increase of the strength of noise. This novel results suggest that an external multiplicative noise sourcecould be used to control gene expression.     

Noise Effects on Oscillator Network of Transcription Regulators

Based on the model describing the regulation of the P_RM operator region of λ phage proposed by Jeff Hasty et al., we study the noise effects on the oscillator network. We find that the additive noise cannot change the period and the amplitude of the relaxation oscillator, but in the multiplicative case, the period of the relaxation oscillator increases to a constant value with the increase of the strength of noise, and the amplitude of the relaxation oscillator also shows increases with the increase of the strength of noise. This novel results suggest that an external multiplicative noise source could be used to control gene expression.     

Phase Noise in Millimeter Wave Phase-Locked Loop with Mixer

In order to estimate the phase noise of millimetre wave (MMW) phase-locked oscillator (PLO), the phase noise relation of signals in MMW phase-locked loop (PLL) with frequency conversion is analyzed. The signals include output of MMW PLO, intermediate frequency (IF) output of harmonic mixer and output of microwave oscillator serving as local oscillator (LO). A method to estimate the phase noise of MMW PLO is presented, which is based on the phase noise of LO and IF. At the same time, a W-band PLO is achieved, and the phase noise values of the three signals are measured. It is shown that the experimental result is well coincident with the analysis of phase noise relation.     

Stability analysis of a noise-induced Hopf bifurcation

We study analytically and numerically the noise-induced transition between an absorbing and an oscillatory state in a Duffing oscillator subject to multiplicative, Gaussian white noise. We show in a non-perturbative manner that a stochastic bifurcation occurs when the Lyapunov exponent of the linearised system becomes positive. We deduce from a simple formula for the Lyapunov exponent the phase diagram of the stochastic Duffing oscillator. The behaviour of physical observables, such as the oscillators mean energy, is studied both close to and far from the bifurcation.Received: 8 August 2003, Published online: 19 November 2003PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.) - 05.45.-a Nonlinear dynamics and nonlinear dynamical systems     

Coherence and stochastic resonance in a birhythmic van der Pol system

We consider the response to uncorrelated noise and harmonic excitation of a birhythmic van der Pol-type oscillator. This system, as opposed to the standard van der Pol oscillator, is characterized by two stable orbits. The noisy oscillator can be analytically mapped, with the technique of stochastic averaging, onto an ordinary bistable system with a bistable (quasi)potential. The birhythmic oscillator can also be numerically characterized through the diagnostics of coherent resonance and the signal-to-noise-ratio. The analysis shows the presence of noise-induced coherent states, influenced by the different time scales of the oscillator.     

白交叉关联色噪声驱动的线性振子的扩散

研究了由白交叉相关的两色噪声所驱动的线性振子的扩散,其中一个噪声受偏置的周期信号所调制.揭示了乘性色噪声的自关联时间τ1和加性色噪声的自关联时间τ2对d-Q,d-D及d-Ω曲线的影响,其中d,Q,D和Ω分别为振子的扩散系数、受信号调制的噪声强度、未受调制的噪声强度和信号的频率.发现d-Q和d-D曲线均存在一个极小值,而d-Ω曲线则在Ω=～Ω处呈现极大值,～Ω为振子频率.推导了使扩散系数d=0的条件.     

OPTIMUM DURATION OF MEASUREMENT FOR REDUCING THERMAL NOISE LIMITATIONS IN TORSION PENDULUM EXPERIMENTS

Statistical estimates of thermal noise fluctuation of a linear harmonic oscillator used to detect weak forces are reexamined with special reference to gravitational experiments. Thermal noise limitation on the measurement of displacement of an oscillator is studied. It is found that the signal response of an oscillator reaches the maximum value while the thermal noise response reaches the minimum value at the same time during the transient state of the oscillator, and the precision of the experiments in the transient state could increase about two orders higher than that in the equilibrium state for typical experiments. It is also found that there is an optimum duration of measurement in which the uncertainty due to thermal noise in this period is the least.     

Ultralow-noise mode-locked laser with coupled optoelectronic oscillator configuration

We describe simultaneous generation of ultralow-noise optical pulses and microwave signal with a mode-locked fiber laser in a coupled optoelectronic oscillator configuration. We demonstrate 9.2-GHz optical and microwave signals with the measured phase noise of -140 dBc/Hz at 10-kHz offset frequency. We show that the mode-locked laser in the photonic oscillator serves as a high-Q filter and is responsible for the observed low phase noise.     

Stability of an Oscillator with Random Mass

We consider an oscillator with a random mass for which the particles of the surrounding medium adhere to the oscillator for some random time after the collision, thereby changing its mass (Brownian motion with adhesion). For the case of dichotomous multiplicative noise, the first moment can diverge, indicating that the system is unable to reach a steady state.     

On the dynamics of a single-bit stochastic-resonance memory device

The increasing capacity of modern computers, driven by Moore’s Law, is accompanied by smaller noise margins and higher error rates. In this paper we propose a memory device, consisting of a ring of two identical overdamped bistable forward-coupled oscillators, which may serve as a building block in a larger scale solution to this problem. We show that such a system is capable of storing a single bit and its performance improves with the addition of noise. The proposed device can be regarded as asynchronous, in the sense that stored information can be retrieved at any time and, after a certain synchronization time, the probability of erroneous retrieval does not depend on which oscillator is being interrogated. We characterize memory persistence time and show it to be maximized for the same noise range that both minimizes the probability of error and ensures synchronization. We also present experimental results for a hard-wired version of the proposed memory, consisting of a loop of two Schmitt triggers. We show that this device is capable of storing a single bit and does so more efficiently in the presence of noise.