Associated relaxation time and intensity correlation function of a bistable system driven by cross-correlation additive and multiplicative coloured noise sources

The associated relaxation time and the intensity correlation function of a bistable system driven by an additive and a multiplicative coloured noise with coloured cross-correlation are investigated. Using the Novikov theorem and the projection operator method, the analytic expressions of the stationary probability distribution P_st(x), the relaxation time T_c, and the normalized correlation function C(s) of the system are obtained. The effects of the noise intensity, the cross-correlation strength λ and the cross-correlation time τ are discussed. By numerical computation, it is found that the cross-correlation strength |λ| and the quantum noise intensity D decrease the relaxation of the system from unstable points. The cross-correlation time τ delays relaxation of the system from unstable points. The cross-correlation strength λ and the cross-correlation time τ can alter the effects of the pump noise intensity Q. Thus, the relaxation time T_c is a stochastic resonant phenomenon, and distribution curves exhibit a single-maximum structure.     

Colored Noise Enhanced Stability in a Tumor Cell Growth System Under Immune Response

In this paper, we investigate a mathematical model for describing the growth of tumor cell under immune response, which is driven by cross-correlation between multiplicative and additive colored noises as well as the nonzero cross-correlation in between. The expression of the mean first-passage time (MFPT) is obtained by virtue of the steepest-descent approximation. It is found: (i) When the noises are negatively cross-correlated (λ<0), then the escape is faster than in the case with no correlation (λ=0); when the noises are positively cross-correlated (λ>0), then the escape is slower than in the case with no correlation. Moreover, in the case of positive cross-correlation, the escape time has a maximum for a certain intensity of one of the noises, i.e., the maximum for MFPT identifies the noise enhanced stability of the cancer state. (ii) The effect of the cross-correlation time τ ₃ on the MFPT is completely opposite for λ>0 and λ<0. (iii) The self-correlation times τ ₁ and τ ₂ of colored noises can enhance stability of the cancer state, while the immune rate β can reduce it.     

Effects of time delay on the statistical fluctuations for a bistable system driven by cross-correlated noises

We study the effects of time delay on the normalized correlation function C(s) and the associated relaxation time T _c for a bistable system with correlations between multiplicative and additive white noises under the condition of small time delay. Using the projection operator method, the expressions of T _c and C(s) are obtained. Based on numerical computations, it is found that the delay time τ slows down the rate of fluctuation decay of dynamical variable for the presence of positive feedback intensity (∈ > 0), while speeds up the rate of fluctuation decay of dynamical variable for the presence of negative feedback intensity (∈ < 0). The effects of the delay time τ on the T _c and C(s) are entirely opposite for ∈ 〉 0 and ∈ < 0.      

The Effects of Time Delay on Stochastic Resonance in a Bistable System with Correlated Noises

The effects of time delay on stochastic resonance (SR) in a bistable system with time delay, correlated noises and periodic signal are studied by using the theory of signal-to-noise ratio (SNR). The expression of the SNR is derived under the adiabatic limit and the small delay time approximation. It is found that: (i) For the case of no correlations between multiplicative and additive noise, the delay time τ can enhance the SNR as a function of the multiplicative noise intensity α and it can restrain the SNR as a function of the additive noise intensity D; (ii) For the case of correlations between multiplicative and additive noise, τ can induce a minimum and maximum in curve of the SNR as a function of α, and can intensively restrain the SNR as a function of the D and there is a critical value of delay tim τ _c =0.1 in the height of the SNR peak with change of τ, i.e., when τ takes value blow τ _c , the τ boosts up the SNR as a function of the strength λ of correlations between multiplicative and additive noise, however, when τ takes value above τ _c , the τ restrains that.     

双色噪声驱动光学双稳系统的弛豫时间研究

研究乘性及加性噪声的自关联时间τ₁和τ₂,交叉关联时间τ对光学双稳系统的弛豫时间的影响.弛豫时间T由Mei等所采用的方法得到. 经数值计算, 结果表明: 两噪声间交叉关联时间τ及乘性噪声的自关联时间τ₁延缓系统涨落的消退(T随τ及τ₁的增大而增大);T随加性噪声的自关联时间τ 关键词： 关联噪声 弛豫时间 光学双稳系统     

Stochastic tumor growth system with two different kinds of time delay

The dynamical properties of a noise-driven tumor cell growth system are investigated when there exist two different kinds of time delays, in the deterministic and fluctuating forces, respectively. Using the approximation probability density approach, the delayed Fokker-Planck equation is obtained. The effects of two different time delays on the stationary probability distribution (SPD), the mean value and the mean passage time (MFPT) are discussed. It is found that the time delay τ₁ in the deterministic force can enhance tumor cell number, while the time delay τ₂ in the fluctuating force can induce a decrease in tumor cell numbers. On the other hand, while τ₁ can hold back the extinction of tumor cells, τ₂ can speed up their extinction.     

Colored Noises-Induced Regime Shifts in a Vegetation Ecological System

In this paper, we investigate the stationary probability distribution and mean first passage time in a vegetation ecological system, which is driven by cross-correlation between intrinsic and extrinsic colored noises as well as the nonzero cross-correlation in between. The impacts of the self-correlation time τ₁ and τ₂, the cross-correlation time τ₃ and intensity k on the stationary probability distribution and mean first passage time are discussed, respectively. Our main results show that: (#em/em#) the self-correlation time τ₁ can induce regime shifts from the desert state to the sustainable vegetation state, while the self-correlation time τ₂, cross-correlation time τ₃ and intensity k can induce regime shifts from the sustainable vegetation state to the desert state; and (ii) the self-correlation time τ₁ or τ₂ can enhance the stability of the sustainable vegetation biomass, while the cross-correlation time τ₃ or strength k weakens the stability of the sustainable vegetation biomass.     

The Asymptotic Limits of Zero Modes of Massless Dirac Operators

Asymptotic behaviors of zero modes of the massless Dirac operator H = α · D + Q(x) are discussed, where α = (α₁, α₂, α₃) is the triple of 4 × 4 Dirac matrices, , and Q(x) = (q _jk (x)) is a 4 × 4 Hermitian matrix-valued function with | q _jk (x) | ≤ C 〈x〉^−ρ, ρ > 1. We shall show that for every zero mode f, the asymptotic limit of |x|² f (x) as |x| → + ∞ exists. The limit is expressed in terms of the Dirac matrices and an integral of Q(x) f (x).      

交叉关联噪声对光学双稳系统平均首通时间的影响

研究了乘性色噪声和加性白噪声驱动的光学双稳系统中噪声对系统两个方向平均首通时间T₊（x_s1→x_s2）和T_-（x_s2→x_s1）的影响（平均首通时间是指从一稳态出发越过势垒到另一稳态所用时间的平均值）.通过Novikov定理和Fox近似方法得到相应的Fokker-Planck方程,利用最速下降法得到T₊（x_s1→x_s2）和T_-（x_s2→x_s1）的表达式.研究发现：乘性噪声强度Q和加性噪声强度D对T₊（x_s1→x_s2）的影响相同,对T_-（x_s2→x_s1）的影响不同;T₊（x_s1→x_s2）随乘性色噪声自关联时间τ的增大而增大,但随噪声间的交叉关联强度λ的增大而减小;T_-（x_s2→x_s1）随τ的增大而减小,随λ的增大而增大. 关键词： 平均首通时间 光学双稳系统 乘性色噪声 加性白噪声     

Colored noise driven systems with inertia

We present a novel approximation scheme, termed unified colored noise approximation (UCNA), for colored Gaussian noise driven nonlinear systems with inertia. This approximation allows one to evaluate static (stationary distributions, moments) as well as dynamical quantities (correlation functions) for small-to-moderate-to-large values of the correlation time. The approximation replaces a three-dimensional Markovian process by a reduced, two-dimensional Markovian dynamics with new drift and diffusion coefficients. For a harmonic potential the stationary moments are reproduced exactly. Most importantly, we present a criterion involving the noise strengthD, the friction strength and the noise color , which describes the region of validity of UCNA in the parameter space given by (D, , ). At small -values we contrast the UCNA with the well-known small approximation. In order to have a comparison onanalytical grounds, we test the static and dynamical predictions of UCNA versus the well-known analytical results obtained from a three-dimensional Ornstein-Uhlenbeck process.     

Stochastic resonance in a bistable system with global delay and two noises

The global time delay is introduced into a bistable system driven by two noises and a periodic signal. The signal power amplification factor η is employed to characterise stochastic resonance (SR) of the system. Numerical simulation results indicate the following. (1) For a periodic signal with low frequency (Ω), the typical behaviour of the SR is lowered monotonically by increasing the delay time τ; for moderate Ω, τ enhances the SR behaviour and then weakens it, with a critical value at which the SR is optimum. (2) Multiplicative noise intensity D and additive noise intensity α have different influences on the SR performance, viz D enhances the SR monotonically while α enforces the SR initially and then restrains it; (3) correlation intensity λ between the two noises always weakens the SR behaviour of the system.     

Influence of collisional coupling on the energy extraction from theB,C, andD state in KrF

Collisional coupling between theB, C, andD states of KrF is measured in a discharge pumped KrF laser. Detection of the time resolved populations in the three states via measurement of the side fluorescence in theB→X, C→A, andD→X bands is used to record the collisional relaxation from theC andD state to theB state during laser emission. The experimentally determined limits for the collisional coupling times are τ_B-C≦1 ns and τ_B-D,τ_C-D<20 ns at total pressures of 2 bar. Investigation of theB→X fluorescence band shows that the vibrational manifold of theB state is not thermalized during laser emission. TheV-V,T coupling time is estimated to be τ_V-V,T=4±2 ns. The influence of collisional coupling on energy extraction in the KrF laser is discussed.     

色高斯噪声驱动双稳系统的多重随机共振研究

研究了由色关联乘性和加性色噪声作用下的双稳系统的随机共振问题,在绝热近似条件下得到了信噪比的表达式.通过分析所得的初始条件为 x(0)=x₊ 时的信噪比,发现了单随机共振和多重随机共振现象;分析了噪声强度、噪声关联时间和关联强度对系统信噪比的影响. 关键词： 多重随机共振 信噪比 双稳模型 色关联色噪声     

Hitting Time and Dimension in Axiom A Systems, Generic Interval Exchanges and an Application to Birkoff Sums

In this note we prove that for equilibrium states of axiom A systems having positive dimension the time τ _B (x) needed for a typical point x to enter for the first time in a typical ball B with radius r behaves for small r as τ _B (x)∼ r ^−d where d is the local dimension of the invariant measure at the center of the ball. A similar relation is proved for a full measure set of interval exchanges. Some applications to Birkoff averages of unbounded (and not L ¹) functions are shown.     

Wavelet analysis of learning and forgetting of photostimulation rhythms for a nonstationary electroencephalogram

Quantitative parameters characterizing transient processes of mastering and forgetting of photostimulation (PST) rhythms for a nonstationary electroencephalogram (EEG) are developed on the basis of a continuous wavelet transformation. Nonstationarity factor K _nst(μ), as well as rhythm mastering K _M (μ) and confinement K _C (μ) factors are calculated for various spectral ranges μ. Photoflash mastering time τ _M = τ _S + τ _I , which is the sum of latent silence period τ _S after PST actuation and the rhythm increasing period τ _I is calculated. In the case of PST, the EEG rhythm retardation time τ _R relative to the beginning of PST is calculated. Rhythm forgetting time τ _F = τ _P + τ _D after PST actuation is the sum of the preservation time τ _P of the corresponding rhythm over a certain time interval and its decay period τ _D . The lag time τ _L of the EEG signal relative to the PST signal after its removal is determined. The proposed method is used in quantitative analysis and classification of transient processes characterizing the properties of the central nervous system. Possible applications of the method in analysis of various nonstationary signals in physics are discussed.     

色噪声驱动的非对称双稳系统的平均首次穿越时间

研究了由关联乘性色噪声及加性白噪声驱动的非对称双稳系统中势阱的非对称性及噪声对系统两个方向平均首次穿越时间的影响. 首先利用一致有色噪声近似推导了系统的稳态概率密度的表达式，根据最速下降法推导了平均首次穿越时间的表达式. 数值结果表明：势阱的非对称性对两个方向的平均首次穿越时间的影响是不同的；由于噪声的关联性，即使对于关联乘性色噪声及加性白噪声驱动的对称双稳系统，两个方向的平均首次穿越时间也不再相等；在lnT₊-r和lnT_{-关键词： 平均首次穿越时间 非对称双稳系统 乘性色噪声 加性白噪声}     

Stationary Properties and Stochastic Resonance for a Saturation Laser Model with Cross-correlation Between Quantum Noise Terms

The stationary properties of a saturation laser model with cross-correlation between the real and imaginary parts of the quantum noise are investigated theoretically. Using the Novikov theorem and the Sargent technique, we obtain the analytic expressions of the stationary probability density distribution, the mean, the variance and the skewness of the saturation laser model. The cross-correlation coefficient λ and other parameters can make the stationary probability density distribution P _st (I) generate interesting two-extrema structure, one-extremum structure, or no-extremum structure. It is clearly found that a first- order-like-transition is induced by the coupling strength |λ| of the complex quantum noise terms in the saturation laser model. When the laser system is operated above the threshold, the mean 〈I〉 becomes larger and the output of the laser intensity increases; however the coupling strength |λ| attenuates the output of the laser intensity. When the laser is operated near and below the threshold, the mean 〈I〉 becomes smaller, the output of the laser intensity decreases, and |λ| still attenuates the output of the laser intensity. When a periodic signal is added to a saturation laser model with cross-correlation between quantum noise terms, the interesting stochastic resonance phenomena occur at λ=0. The noise intensity Q decreases the values of the resonance peak, however, the amplitude of the periodic signal B enhances the values of the resonance peak.     

On the perturbative stability of the QCD predictions for the ratio R=FL/FT in heavy-quark leptoproduction

We analyze the perturbative and parametric stability of the QCD predictions for the Callan–Gross ratio, R(x,Q ²)=F _L /F _T , in heavy-quark leptoproduction. We consider the radiative corrections to the dominant photon–gluon fusion mechanism. In various kinematic regions, the following contributions are investigated: exact NLO results at low and moderate Q ²≲m ², asymptotic NLO predictions at high Q ²≫m ², and both NLO and NNLO soft-gluon (or threshold) corrections at large Bjorken variable x. Our analysis shows that large radiative corrections to the structure functions F _T (x,Q ²) and F _L (x,Q ²) cancel each other in their ratio R(x,Q ²) with good accuracy. As a result, the NLO contributions to the Callan–Gross ratio are less than 10% in a wide region of the variables x and Q ². We provide compact LO predictions for R(x,Q ²) in the case of low x ≪1. A simple formula connecting the high-energy behavior of the Callan–Gross ratio and low-x asymptotics of the gluon density is derived. It is shown that the obtained hadron-level predictions for R(x→0,Q ²) are stable under the DGLAP evolution of the gluon distribution function. Our analytic results simplify the extraction of the structure functions F ₂ ^c (x,Q ²) and F ₂ ^b (x,Q ²) from measurements of the corresponding reduced cross sections, in particular at DESY HERA.     

Observation of a correlation between the dependences of the radiative decay time and intensity of the impurity core-valence luminescence on excitation energy in K1−x CsxCl and Rb1−x CsxCl mixed crystals

A study of the spectral and kinetic parameters of impurity core-valence luminescence (CVL) excited by synchrotron radiation in K_1−x Cs_xCl and Rb_1−x Cs_xCl mixed crystals is reported. The dependences of the intensity and radiative decay time τ of impurity CVL on exciting photon energy hν have been found to be correlated. It is concluded that the observed τ = f(hv) relation originates from nonradiative decay of core excitations near the surface, which govern also the excitation spectral features in this spectral region. Fiz. Tverd. Tela (St. Petersburg) 41, 1973–1975 (November 1999)     

Signature of chaos in power spectrum

We investigate the nature of the numerically computed power spectral densityP(f, N, τ) of a discrete (sampling time interval,τ) and finite (length,N) scalar time series extracted from a continuous time chaotic dynamical system. We highlight howP(f, N, τ) differs from the true power spectrum and from the power spectrum of a general stochastic process. Non-zeroτ leads to aliasing;P(f, N, τ) decays at high frequencies as [πτ/sinπτf]², which is an aliased form of the 1/f ² decay. This power law tail seems to be a characteristic feature of all continuous time dynamical systems, chaotic or otherwise. Also the tail vanishes in the limit ofN → ∞, implying that the true power spectral density must be band width limited. In striking contrast the power spectrum of a stochastic process is dominated by a term independent of the length of the time series at all frequencies.