The maximal Lyapunov exponent of a co-dimension two-bifurcation system excited by a bounded noise

In the present paper,the maximal Lyapunov exponent is investigated for a co-dimension two bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by a bounded noise.By using a perturbation method,the expressions of the invariant measure of a one-dimensional phase diffusion process are obtained for three cases,in which different forms of the matrix B,that is included in the noise excitation term,are assumed and then,as a result,all the three kinds of singular boundaries for one-dimensional phase diffusion process are analyzed.Via Monte-Carlo simulation,we find that the analytical expressions of the invariant measures meet well the numerical ones.And furthermore,the P-bifurcation behaviors are investigated for the one-dimensional phase diffusion process.Finally,for the three cases of singular boundaries for one-dimensional phase diffusion process,analytical expressions of the maximal Lyapunov exponent are presented for the stochastic bifurcation system.     

一类三维随机动力系统的最大Lyapunov指数

基于一维扩散过程的奇异边界理论,使用摄动方法研究了白噪声参激的一类余维二分岔系统的最大Lyapunov指数渐近表达式和数值解,主要讨论了一维相扩散过程同时存在两类奇异边界以及FPK方程存在平稳解的一般性条件. 通过对参激噪声作用项系数矩阵的分析,给出了不变测度的解析解及其相应的Monte Carlo数值仿真结果,并导出了一维相扩散过程P分岔点的确定方法. 对于一类特殊情形,给出了最大Lyapunov指数的渐近表达式;对于参激噪声作用项系数矩阵的一般情形,则给出了系统最大Lyapunov指数的数值结果.      

The Lyapunov exponent for a codimension two bifurcation system that is driven by a real noise

For a co-dimension two bifurcation system on a three-dimensional central manifold, which is parametrically excited by a real noise, a model of enhanced generality is developed in the present paper by assuming the real noise to be a component of the output of a linear filter system—a zero-mean stationary Gaussian diffusion vectoral process, which conforms with the detailed balance condition. The strong mixing condition is removed in the present paper. To handle the complexities encountered in the present work, an asymptotic analysis approach and the eigenfunction expansion of the solution to the relevant FPK equation are employed in the construction of the asymptotic expansions of the invariant measure and the maximal Lyapunov exponents for the relevant system.     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO_DIMENSION TWO BIFURCATION SYSTEM (Ⅰ)

Ｐｒｅｓｅｎｔｌｙ,ｔｈｅｃｏｍｐｕｔａｔｉｏｎａｌａｐｐｒｏａｃｈｔｏｍａｘｉｍａｌＬｙａｐｕｎｏｖｅｘｐｏｎｅｎｔｓｆｏｒｎｏｎｌｉｎｅａｒｓｔｏｃｈａｓｔｉｃｓｙｓｔｅｍｓｈａｓｅｍｅｒｇｅｄａｓｏｎｅｐｒｉｍａｒｙｆｏｃｉｏｆｒｅｓｅａｒｃｈｉｎｔｅｒｅｓｔｉｎｔｈｅｆｉｅｌｄｏｆｓｙｓｔｅｍｓｔａｂｉｌｉｔｙａｎｄｓｔｏｃｈａｓｔｉｃｂｉｆｕｒｃａｔｉｏｎ．ＴｈｉｓｉｓｍａｉｎｌｙａｔｔｒｉｂｕｔｅｄｔｏｔｈｅｆａｃｔｔｈａｔｍａｘｉｍａｌＬｙａｐｕｎｏｖｅｘｐｏｎｅｎｔｈａｓｂｅｅｎ…     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅰ)

For a real noise parametrically excited co-dimension two bifurcation system on three-dimensional central manifold, a model of enhanced generality is developed in the present paper by assuming the real noise to be an output of a linear filter system, namely, a zero-mean stationary Gaussian diffusion process that satisfies the detailed balance condition. On such basis, asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are established by use of Arnold asymptotic analysis approach in parallel with the eigenvalue spectrum of Fokker-Planck operator. Foundation item: the National Natural Science Foundation of China (19602016)     

Soft and Hard Wall in a Stochastic Reaction Diffusion Equation

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a “soft” repulsion from the boundary. We finally show how a “hard” repulsion can be obtained by an extra diffusive scaling.     

Morris-Lecar系统中通道噪声诱导的自发性动作电位研究

在生物物理学中, 越来越多的现象是由于分段确定性的动力系统与连续时间马氏过程之间的耦合作用而产生的. 因为这种耦合性, 相关的数学模型更适合取为随机混合系统而不是扩散过程(基于It?随机微分方程). 本文从理论上和数值上研究了在弱噪声条件下无鞍点状态的随机混合Morris-Lecar系统中, 由通道噪声诱导的自发性放电现象. 一个动作电位的初始阶段可视为噪声诱导的逃逸事件, 其最优路径和拟势可由辅助Hamilton系统给出. 由于系统不存在鞍点, 因此可选择虚拟分界线(ghost separatrix)为阈值, 研究噪声诱导的自静息态的逃逸事件. 通过计算在阈值处的拟势, 便可发现其值有一个明显的最小值, 其作用类似于鞍点. 通过改进的Monte Carlo模拟方法, 计算了历程概率分布, 其结果对初始阶段和兴奋阶段的理论解均给出了验证. 此外, 基于前人将拟势等高线作为阈值的另一种选择, 我们对两种阈值取法的优劣性进行了比较. 最后, 本文研究了钠离子和钾离子通道噪声的不同组合对最优路径和拟势的影响. 结果表明: 钾离子通道噪声在自发性放电过程中起主导作用, 且两种噪声强度存在一个最优比例能使总的噪声强度达到最小.      

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅱ)

For a co-dimension two bifurcation system on a three-dimensional central manifold, which is parametrically excited by a real noise, a rather general model is obtained by assuming that the real noise is an output of a linear filter system-a zeromean stationary Gaussian diffusion process which satisfies detailed balance condition. By means of the asymptotic analysis approach given by L. Arnold and the expression of the eigenvalue spectrum of Fokker-Planck operator, the asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are obtained. Foundation item: the National Natural Science Foundation of China (19602016)     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅱ)

ＴｈｅｄｅｃｉｓｉｖｅｅｆｆｅｃｔｓｏｆｓｉｎｇｕｌａｒｐｏｉｎｔｓａｎｄｂｏｕｎｄａｒｉｅｓｂｅｌｏｎｇｉｎｇｔｏｓｏｌｕｔｉｏｎｐｒｏｃｅｓｓｅｓｏｆＩｔｏｓｔｏｃｈａｓｔｉｃｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｏｎｔｈｅｓｉｔｕａｔｉｏｎｓｏｆｄｉｓｔｒｉｂｕｔｉｏｎｓｏｆｅｒｇｏｄｉｃｃｏｍｐｏｎｅｎｔｓ,ｔｈｅｅｘｉｓｔｅｎｃｅａｎｄｔｈｅｆｏｒｍｓｏｆｉｎｖａｒｉａｂｌｅｍｅａｓｕｒｅｓ,ｔｈｅｓａｍｐｌｅｐｒｏｐｅｒｔｉｅｓ（ｉｎｃｌｕｄｉｎｇｓａｍｐｌｅｓｔａｂｉｌｉｔｙ）…     

Inclusion of Preferential Diffusion in Simulations of Premixed Combustion of Hydrogen/Methane Mixtures with Flamelet Generated Manifolds

In this paper we study the possibility to account for preferential diffusion effects in lean turbulent premixed flames in numerical predictions with reduced chemistry. We studied the situation when hydrogen is added to methane at levels of 20% and 40% by volume in the fuel, at lean combustion (??=?0.7) with air. The base case of pure methane was used as a reference. In this case preferential diffusion effects are negligible. First the sensitivity of the mass burning rate to flame stretch was investigated, in one dimensional computations with detailed chemistry, to set reference values. Then the framework of the Flamelet Generated Manifolds (FGM) was used to construct an adequate chemical method to take preferential diffusion into account, without the need for using detailed chemistry. To that end a generalization of the method was presented in which five controlling variables are required. For this system, proper transport equations and effective Lewis numbers where derived. In practice not all five variables are necessary to include and as a first step we limited the amount in the numerical tests in this study to two controlling variables. The method was then tested in configurations in which there was an interaction of coherent vortices and turbulence with flames. It was demonstrated that a minimum of two controlling variables is needed to account for the changed mass burning rate as function of stretch and curvature. It was shown that one-dimensional FGM as well as one-step Arrhenius kinetics can not describe this relation.     

Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation

The stochastic averaging method for strongly non-linear oscillators with lightly fractional derivative damping of order α (0<α≤1) subject to bounded noise excitations is proposed by using the generalized harmonic function. The system state is approximated by a two-dimensional time-homogeneous diffusion Markov process of amplitude and phase difference using the proposed stochastic averaging method. The approximate stationary probability density of response is obtained by solving the reduced Fokker–Planck–Kolmogorov (FPK) equation using the finite difference method and successive over relaxation method. A Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary resonance, the stochastic jump of the Duffing oscillator with fractional derivative damping and its P-bifurcation as the system parameters change are examined for the first time using the stationary probability density of amplitude.     

Spatio-temporal sine-Wiener bounded noise and its effect on Ginzburg–Landau model

In this work we introduce a spatio-temporal bounded noise derived by the sine-Wiener noise and by the spatially colored unbounded noise proposed by García-Ojalvo, Sancho, and Ramírez-Piscina (GSR noise). We characterize the behavior of the equilibrium distribution of this novel noise by showing its dependence on both the temporal and the spatial autocorrelation lengths. In particular, we show that the distribution experiences a stochastic transition from bimodality to trimodality. Then, we employ the noise here defined to study the emergence of phase transitions in the real Ginzburg–Landau model. Various phenomena are evidenced by means of numerical simulations, among which reentrant transitions, as well as differences in the response of the system to “equivalent” GSR additive noise perturbations.     

Stochastic averaging method for estimating first-passage statistics of stochastically excited Duffing–Rayleigh–Mathieu system

The first-passage statistics of Duffing-Rayleigh- Mathieu system under wide-band colored noise excitations is studied by using stochastic averaging method. The motion equation of the original system is transformed into two time homogeneous diffusion Markovian processes of amplitude and phase after stochastic averaging. The diffusion process method for first-passage problem is used and the corresponding backward Kolmogorov equation and Pontryagin equation are constructed and solved to yield the conditional reliability function and mean first-passage time with suitable initial and boundary conditions. The analytical results are confirmed by Monte Carlo simulation.     

Quadratic and rate-independent limits for a large-deviations functional

We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth–death process on a lattice, with rates derived from Kramers’ law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits, we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via ‘L log L’ gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process.     

Study of a new macro-particle model for the low-temperature pyrolysis of dried wood chips

 This paper presents a transient one-dimensional mathematical model which simulates the pyrolysis of a single dried wood particle. The porous wood particle is considered as a two-phase system: the solid phase consisting of wood and char and the gas phase consisting of volatiles and tar. Conservation equations for mass, momentum and energy are developed for each phase. Chemical processes are described through an existing one-stage three-reactions scheme, leading separately to char, tar and volatiles evolution dynamics. The variation of transport and physical properties with temperature and with composition is presented by algebraic equations. The model presented in this paper is more advanced than the current models found in literature, since it contains physical effects not included in past models, such as cross diffusion, differing solid and gas phase temperatures and a transient gas phase momentum equation incorporating the wall friction experienced by a fluid flowing through a porous medium. Furthermore, an adequate reference system for enthalpy, based on temperature dependent reaction heats, is used. The mathematical equations with initial and boundary conditions are solved numerically by means of a commercial CFD code (PHOENICS). The validity of the pyrolysis kinetics scheme is examined through comparison with experimental data. Furthermore, the macro-particle model is validated by (1) examining the limitations and importance of the newly-modelled effects (different solid phase and gas phase temperature, cross diffusion and wall friction) and (2) making a comparison between predicted and experimental results for large particles. Received on 18 December 2000     

Application of a n-Phase Model to the Diffusion Coefficient of Chloride in Mortar

The determination of the chloride diffusion coefficient of a concrete is needed to help the prediction of the service life of concrete structure. In this paper, we propose first a critical review of models for chloride diffusion coefficients already used in literature at different scales and then we develop an analytical model, which takes into account the characteristics of the different phases of concrete. These materials are treated as a three-phase composite, consisting of a cement continuous phase, of an aggregates dispersed phase and of an interface transition zone. Chloride diffusion coefficient using an n-layered inclusion-based micromechanical modeling is predicted. The details of calculations are summarized hereafter and experimental measurements obtained on mortars are compared with predicted results.     

白噪声参激余维2分叉系统研究

为考查白噪声参激的具有非半简双零特征值的一类余维二分叉系统的样本稳定性并确定其首次分叉点的位置，本文使用Ｌ．Ａｒｎｏｌｄ的渐近分析方法研究了系统最大Ｌｙａｐｕｎｏｖ指数的渐近分析式．为进一步研究噪声对于此系统退化分叉的影响，本文将使用一维扩散过程的奇异边界理论，考查“隐藏在余维２分叉点之后”同宿分叉系统受参激白噪声影响的分叉行为．     

Stochastic Hopf bifurcation of quasi-nonintegrable-Hamiltonian systems

A new procedure for analyzing the stochastic Hopf bifurcation of quasi-non-integrable-Hamiltonian systems is proposed. A quasi-non-integrable-Hamiltonian system is first reduced to an one-dimensional Itô stochastic differential equation for the averaged Hamiltonian by using the stochastic averaging method for quasi-non-integrable-Hamiltonian systems. Then the relationship between the qualitative behavior of the stationary probability density of the averaged Hamiltonian and the sample behaviors of the one-dimensional diffusion process of the averaged Hamiltonian near the two boundaries is established. Thus, the stochastic Hopf bifurcation of the original system is determined approximately by examining the sample behaviors of the averaged Hamiltonian near the two boundaries. Two examples are given to illustrate and test the proposed procedure.     

带冷却气流的亥姆霍兹共振器的声类比模型

亥姆霍兹共振器(HR)作为典型的被动消声装置,常被安装于航空发动机和燃气轮机的燃烧室上用以吸收噪声进而抑制燃烧热声振荡.在实际应用中,为防止燃烧室内高温气体损坏HR,常引入冷却气流从HR的背部空腔通过其颈部流入燃烧室,以保护HR.该冷却气流的温度一般显著低于燃烧室内的燃气温度.将这样的HR安装到燃烧室上时,该温差可能影...     

一种通过条纹方向图提取散斑条纹中心线的方法

在光测力学中,如何从单幅散斑干涉条纹图中消除散斑噪声,高精度自动提取相位信息,是一项较困难的工作。本文提出了一种计算条纹方向的简单算法,以及通过条纹方向图提取散斑条纹中心线的方法。实验表明此方法对高噪声的散斑条纹图非常有效。