SIMILARITY REDUCTIONS FOR THE NONLINEAR EVOLUTION EQUATION ARISING IN THE FERMI-PASTA-ULAM PROBLEM

ＩｎｔｒｏｄｕｃｔｉｏｎＦｅｒｍｉ,ＰａｓｔａａｎｄＵｌａｍ[1]ｐｒｅｓｅｎｔｅｄａｗｅｌｌ＿ｋｎｏｗｎｐｒｏｂｌｅｍｗｈｙｓｏｌｉｄｓｈａｄｆｉｎｉｔｅｈｅａｔｃｏｎｄｕｃｔｉｖｉｔｙ ,ｗｈｉｃｈｗａｓｃａｌｌｅｄｔｈｅＦｅｒｍｉ＿Ｐａｓｔａ＿Ｕｌａｍ(ＦＰＵ)ｐｒｏｂｌｅｍ .Ｔｈｅｓｏｌｉｄｉｓｍｏｄｅｌｌｅｄｂｙａｏｎｅ＿ｄｉｍｅｎｓｉｏｎａｌｌａｔｔｉｃｅ ,ａｓｅｔｏｆｍａｓｓｅｓｃｏｕｐｌｅｄｂｙｓｔｒｉｎｇ .Ｔｈｅｅｑｕａｔｉｏｎｓｇｏｖｅｒｎｉｎｇｔｈｅｄｙｎａｍｉｃｓｏｆｔｈｉｓｌａ…     

THE EXISTENCE OF PERIODIC SOLUTION OF THE FOURTH ORDINARY NONLINEAR DIFFERENTIAL EQUATION CAUSED BY FLOW－IN

ＴＨＥＥＸＩＳＴＥＮＣＥＯＦＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＯＦＴＨＥＦＯＵＲＴＨＯＲＤＩＮＡＲＹＮＯＮＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＣＡＵＳＥＤＢＹＦＬＯＷ－ＩＮＤＵＣＥＤＶＩＢＲＡＴＩＯＮＧｕＱｉｎｇ－ｆａｎｇ（顾清芳）Ｔａ...     

A PERTURBATION SOLUTION IN THE NONLINEAR THEORY OF CIRCULAR PLATES

In this paper,we analyzed some problems of non-linear circular plates by means of perturbation me-thod.The perturbation parameters chosen here areobtained from solving the equations and are not cer-tain mechanical quantities given precedently.Thisis an extension of W.Z.Chien’s perturbation method,Which uses the central deflection as the perturba-tion parameter.     

ON THE REPRESENTATION OF FINITE ROTATION IN NONLINEAR FIELD THEORY OF CONTINUUM MECHANICS

One of the important basic theoretical problems in the development of continuummechanics is the separation of finite strain and finite rotation at a point in the displacementfield.Now it is certain that S-R decomposition theorem provides a rational theoreticalsolution for this problem.The purpose of this paper is to clarify some misleading basicconcepts of finite rotation of deformed body in current literature,and to promote furtherprogress.     

ON LIAPOUNOFF’S METHOD IN THE THEORY OF STABILITY OF LAMINAR FLUID FLOWS

In[1]Zhou extended some Liapounoff‘s theorems of the theory of stability in the case of plane laminar fluid flows.In[2]Zhou and Li investigated the eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation,and obtained some new results.In this paper,based on the results of[1]and[2]we shall prove:(1)For the linearized problem the definition of stability according to the eigenvalues of Orr-Sommerfeld equation and that according to the perturbation.energy are equivalent;(2)The method of linearization is admissible for the stability pro-blem of plane laminar fluid flows for sufficiently small initial disturbance.     

EXTENDED VARIATIONAL PRINCIPLE IN NONLINEAR THEORY OF ELASTICITY

From the extended energy functional, a unified variational principle and an extended variational principle are given and proved in this paper. It is about the non-conservative, abrupt and divided domains static (or kinetic) system and by using the undetermined energy functional, we can immediately deduce various variational principles.     

ON THE STABILITY OF FORCED DISSIPATIVE NONLINEAR SYSTE

ＯＮＴＨＥＳＴＡＢＩＬＩＴＹＯＦＦＯＲＣＥＤＤＩＳＳＩＰＡＴＩＶＥＮＯＮＬＩＮＥＡＲＳＹＳＴＥＭ￥ＣｈｅｎＤａｄｕａｎ（陈达段）ＬｉｕＸｉａｏｍｉｎｇ（刘晓明）ＳｈｉＷｅｉｈｕｉ（施惟慧）（ＳｈａｎｇｈａｉＵｎｉｖｅｒｓｉｔｙ．Ｓｈａｎｇｈａｉ２００...     

THE LINEARIZATION OF CERTAIN CLASS OF NONLINEAR SIMULTANEOUS EQUATION SET CONTAINING AMPLITUDE AND PHASE FREQUENCY CHARACTERISTICS AND APPLICATION

A class of complex function of rational fraction type G(jω)=[1 a_1jω a_2(jω)~2 … a_m(jω)~n]/[b_0 b_1jω b_2(jω)~2 … b_n(jω)~n ] is frequently used to describe the dyna-mical properties of systems.It is however quite difficult to es-tablish a mathematical model of this type on the basis of ampli-tude and phase frequency data collected from experiments conductedon the related physical system.Since the erection of mathematicalmodel G(jω)would involve the solution of a set of nonlinear si-multaneous equations|G(jω_i)|=g_i∠G (jω_i)=θ_i i=1, 2,…,r. with the unknown coefficients a_isand b_is(i=0,1,…,m,…,n)in G(jω).Up to now,these nonlinear equa-tions have been considered to be very difficult to solve directly.In spite of the fact there are special computer programmes in cer-tain software packages available to tackle this problem,it is byno means an easy task due to the complex procedures involved inpicking up a set of initial values that should be close enough tothe exact solutions.This paper     

A CRITERION FOR THE STABILITY OF MOTION OF NONLINEAR SYSTEM

As to an autonomous nonlinear system, the stability of the equilibrium state in a sufficiently small neighborhood of the equilibrium state can be determined by eigenvalues of the linear part of the nonlinear system provided that the eigenvalues are not in a critical case. Many methods may be used to detect the stability for a linear system. A lot of researches for determining the stability of a nonlinear system are completed by mathematicians and mechanicians but most of them are methods for the special forms of nonlinear systems. Till now, none of these methods can be conveniently applied to all nonlinear systems. The method introduced by this paper gives the necessary and sufficient conditions of the stability of a nonlinear system. The familiar Krasovski's method is a special case of this method     

THE EQUATION OF MOTION FOR THE SYSTEM OF THE VARIABLE MASS IN THE NON－LINEAR NON－HOLONOMIC SPACE

ＴＨＥＥＱＵＡＴＩＯＮＯＦＭＯＴＩＯＮＦＯＲＴＨＥＳＹＳＴＥＭＯＦＴＨＥＶＡＲＩＡＢＬＥ　ＡＳＳＩＮＴＨＥＮＯＮ－ＬＩＮＥＡＲＮＯＮ－ＨＯＬＯＮＯＭＩＣＳＰＡＣＥＴＨＥＥＱＵＡＴＩＯＮＯＦＭＯＴＩＯＮＦＯＲＴＨＥＳＹＳＴＥＭＯＦＴＨＥＶＡＲＩＡＢＬＥ...     

The re-examination of the weakly nonlinear theory of hydrodynamic stability

The weakly nonlinear theory has been widely applied in the problem of hydrodynamicstability and also in other fields.However,although its application has been successful forsome problems,yet,for other problems,the results obtained are not satisfactory,especiallyfor problems like transition or the evolution of the vortex in the free shear flow,for whichthe goal of the theoretical investigation is not seeking for a steady state,but predicting anevolutional process.In this paper,we shall examine the reason for the unsuccessfulness andsuggest ways for its amendment.     

On problems in the weakly nonlinear theory of hydrodynamic stability and its improvement

There are three main problems in the weakly nonlinear theory of hydrodynamic stability: (1)The radius of convergence with respect to the perturbation parameter is too small and there is no concrete estimation for it. (2)The solution has a special structure, thus in general, it can not satisfy the initial condition posed by many practical problems. (3)When the linear part of its solution does not correspond to a neutral case, there are more than one way in determining the Landau constants, and practically no one knows which is the best way. In this paper, problems(1) and (2)are solved theoretically, and ways for its improvement have been proposed. By comparing the theoretical results with those obtained by numerical simulations, problem(3)has also been clarified. Project supported by the National Natural Science Foundation of China     

On the energy equation for a spatially developing instability wave in the theory of hydrodynamic stability

In this article, it is shown that the energy equation for a spatially developing disturbance used in all the literatures dealing with the problem of hydrodynamic stability suffers from a small, but crucial error.     

The linearization of certain class of nonlinear simultaneous equation set containing amplitude and phase frequency characteristics and application

A class of complex function of rational fraction type is frequently used to describe the dynamical properties of systems. It is however quite difficult to establish a mathematical model of this type on the basis of amplitude and phase frequency data collected from experiments conducted on the related physical system. Since the erection of mathematical model G(j) would involve the solution of a set of nonlinear simultaneous equations with the unknown coefficients a_is and b_is(i=0, 1, ..., m, ..., n) in G(j). Up to now, these nonlinear equations have been considered to be very difficult to solve directly. In spite of the fact there are special computer programmes in certain software packages available to tackle this problem, it is by no means an easy task due to the complex procedures involved in picking up a set of initial values that should be close enough to the exact solutions. This paper proposes a simplified method of linearizing these nonlinear equations set so that direct solution is possible. The method can also be applied to systems with factors of (j) andej0 in G(j). An illustration by a workable example is furnished at the end of this paper to show its versatility.     

超空泡航行器升沉运动流体动力特性研究

基于Rayleigh—Plesset方程的单一介质可变密度Mixture模型,采用动坐标系法完成了对带空泡航行器作升沉运动时空化流场的数值模拟,并分析了该状态下航行器流体动力特性变化规律,对升沉运动频率和幅值对航行器流体动力系数的影响以及升沉运动对航行器自主运动弹道的影响进行了对比研究。研究结果表明,升沉运动过程中,航行器流体动力系数出现周期性变化,增加升沉运动频率和幅值不仅将导致航行阻力和升力增大,同时会造成阻力系数和升力系数幅值波动加剧;升沉运动使航行器速度及角速度出现周期性震荡。本文结论为研究超空泡航行器动力学建模、附加质量计算及弹道设计奠定了基础。     

On the theory of stability of a cylindrical hydrodynamic suspension

The stability of a cylindrical hydrodynamic suspension is investigated with complete account for the dependence of the fluid velocity distribution profile on the radial coordinate in the supporting layer without an a priori representation of the inertial terms in the fluid layer in terms of the velocity averaged over the clearance cross-section. It is shown that the central position of a light inner body in the cylindrical hydrodynamic suspension is asymptotically stable. The suspension remains asymptotically stable for a fairly long time as the light inner body is displaced from the central position along the dynamic equilibrium curve. The central position of a heavy inner body is unstable; however, as it is displaced from the central position under the action of a constant external decentering force a stability domain develops.     

The stability of difference schemes of a higher dimensional parabolic equation

This paper proposes a new method to improve the stability condition of differencescheme of a parabolic equation.Necessary and sufficient conditions of the stability of thisnew method are given and proved.Some numerical examples show that this method hassome calculation advantages.     

The asymptotic stability analysis in stochastic logistic model with Poisson growth coefficient

The asymptotic stability of a discrete logistic model with random growth coefficient is studied in this paper. Firstly, the discrete logistic model with random growth coefficient is built and reduced into its deterministic equivalent system by orthogonal polynomial approximation. Then, the linear stability theory and the Jury criterion of nonlinear deterministic discrete systems are applied to the equivalent one. At last, by mathematical analysis, we find that the parameter interval for asymptotic stability of nontrivial equilibrium in stochastic logistic system gets smaller as the random intensity or statistical parameters of random variable is increased and the random parameter's influence on asymptotic stability in stochastic logistic system becomes prominent.     

On the stability of the solutions to an integral evolution equation of the non linear particle transport theory

Summary In this paper we study the stability, with respect to the initial data, of the solutions to a non linear integral equation arising in particles transport theory.

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Sommario In questo lavoro si prova la stabilità, rispetto ai dati iniziali, delle soluzioni di un'equazione integrale non lineare che si incontra in teoria del trasporto di particelle.