What are the separatrix values named by Leontovich on homoclinic bifurcation

Introduction LetVλ(x),x∈R2,λ∈RkbeafamilyofplanarC∞(oranalytic)vectorfields.Suppose thatforλ=λ0,Vλ0hasahyperbolicsaddlepointattheorigin(0,0)inthephaseplaneandthere isahomoclinicorbit(oraseparatrixloop)Γ0totheorigin.Thehyperbolicityratioattheorigin isr(λ)=-λ1/λ2,whereλ1<0<λ2arethetwoeigenvaluesofthelinearizedsystematthe origin.Generally,whenλ≠λ0and｜λ-λ0｜issmallenough,nearΓ0limitcycleswillbe created.Thisso_calledhomoclinicbifurcationhasbeenstudiedbyalotofauthors[1-3].Deno…     

关于奇异摄动理论在自引力介质波动中的应用

本文通过自引力介质中波动的两个具体实例,讨论了与应用奇异摄动法有关的一些问题。第一个实例是密度波理论中遇到的带有三个迴转点(其中两个相距很近)的复本征值问题,指出:由于此问题中迴转点分布的几何结构敏感地依赖于复本征值ω,因此,即使为了求得初级近似的一致有效渐近解,也必须事先对于复本征值ω对摄动参数λ的依赖性作某些明显的分析:将ω(λ)展成级数sum from l=0 to ∞ ω_jλ~(-i),并根据问题中的辐射条件,估计出ω的实、虚部关于λ的不同量阶。这是此问题分析过程中的关键性步骤。这一作法在形式上与通常的变形参数法(Lindstedt-Poincaré方法)有些类似,但实质并不相同。第二个实例是自引力介质中弱非线性波的渐近周期解及色散关系。指出:倘若采用展开式ω=ω_o(k) εω_1(k) ε~2ω_2(k) ……(其中ω为频率,k为波数),则所得弱非线性色散关系当λ→λ_J(Jeans波长)时不一致有效。本文在分析了产生这种不一致有效性原因的基础上,改用展开式k=k_o εk_1(ω) ε~2k_2(ω) …。于是,不但弱非线性渐近解关于(x,t)一致有效(当波长λ固定),而且弱非线性色散关系关于波长λ也一致有效。从而指出在此问题以及其它类似问题中根据物理特点适当选取变形参数的必要性。     

HIGHER ORDER MULTIVARIABLE NRLUND EULER-BERNOULLI POLYNOMIALS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｋｔｈｏｒｄｅｒＮ ｒｌｕｎｄＥｕｌｅｒｐｏｌｙｎｏｍｉａｌｓＥ(ｋ)ｖ (ｘ｜ω1,… ,ωｋ)ａｎｄＢｅｒｎｏｕｌｌｉｐｏｌｙｎｏｍｉａｌｓＢ(ｋ)ｖ (ｘ｜ω1,… ,ωｋ)ａｒｅｇｉｖｅｎｂｙｔｈｅｆｏｌｌｏｗｉｎｇｅｘｐａｎｓｉｏｎｆｏｒｍｕｌａｒｅｓｐｅｃｔｉｖｅｌｙ[1]2 ｋｅｘｔ(ｅω1 ｔ+ 1 )… (ｅωｋｔ+ 1 ) = ∞ｖ=0Ｅ(ｋ)ｖ (ｘ｜ω1,… ,ωｋ) ｔｖｖ!,( 1 )ω1…ωｋｔｋｅｘｔ(ｅω1 ｔ-1 )… (ｅωｋｔ-1 ) = ∞ｖ=0Ｂ(ｋ)ｖ (ｘ｜ω1,… ,ωｋ) ｔｖｖ!. ( 2 )Ｃｌｅａｒｌｙｔ…     

BOUNDARY INTEGRAL FORMULA OF ELASTIC PROBLEMS IN CIRCLE PLANE

IntroductionConcerningtheelasticplaneprobleminaunitcircle ,ZhengShenzhouandZhengXueliangdevelopedaboundaryintegralformulaofthestressfunction[1]:Φ(r,θ) =-( 1 -r2 ) 24π ∫2π0ν( φ)1 -2rcos(θ-φ) r2 dφ　　 12π∫2π011 -2rcos(θ-ω) r2 dω∫2π0μ( φ)1 -cos(ω-φ) dφ　　 1 -r22π∫2π0μ( φ)1 -2rcos(θ -φ) r2 dφ　　 ( 0 ≤r <1 ) ,( 1 )whereμ(θ) =Φ(r,θ) ｜r=1,ν(θ) = Φ n r=1= Φ r r=1.Intheformula ( 1 )theseconditemisastrongsingularintegral,itshouldbeunderstoodasanintegra…     

DYNAMICAL CHARACTER FOR A PERTURBED COUPLED NONLINEAR SCHRDINGER SYSTEM

Introduction WeconsidertheperturbedcouplednonlinearSchr dingerequations(CNLS)iq1t=q1xx 2[|q1|2 |q2|2-ω2]q1 iε(^D1q1-r1),iq2t=q2xx 2[|q1|2 |q2|2-ω2]q2 iε(^D2q2-r2),(1)whereqjis2πperiodicandeveninx.rjisconstant,^Djisboundeddissipativeoperatorand assumedtotaketheform^Djqj=-αjqj βj^Bqj,(2)αjandβjarepositiveconstants,j=1,2.^BisaFouriertruncationofLaplaceoperator xx,i.e.,^Bcos(kx)=-k2cos(kx),k0isasmallperturbation parameter.Th…     

Uniqueness for the solutions of elastic thin plates and shallow shells as well as their contact problem with half space

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｇｏｖｅｒｎｉｎｇｅｑｕａｔｉｏｎｓｆｏｒａｎｉｎｖｅｒｔｅｄｅｌａｓｔｉｃｔｈｉｎｓｈａｌｌｏｗｓｈｅｌｌｏｎｅｌａｓｔｉｃｈａｆｓｐａｃｅｉｎｓｔａｔｉｃｂｅｎｄｉｎｇａｒｅ[1]4-Ｅｉｈｉ2ｋｗ=02=2/ｘ2 2/ｙ2,2ｋ=ｋｙ(2/ｘ2) ｋｘ(2/ｙ2),(1)Ｄｉ4ｗ 2ｋ=ｑ-ｐ,(2)ｗ(ｘ,ｙ)=λｅ 2Ｇｅ4πＧｅ(λｅ Ｇｅ)∫∫ｓｐ(ξ,η)ｄξｄη(ｘ-ξ)2 (ｙ-η)2,(3)ｗｈｅｒｅｉｎＥｑｓ.(1),(2)ａｎｄ(3),ｗ(ｘ,ｙ)ｉｓｔｈｅｄｅｆｌｅｃｔｉｏｎｏｆｔｈｅｓｈｅｌｌ,(ｘ,ｙ)ｉｓｔｈｅｍ…     

小问题

201.一平面机构(图1)(?)已知 OA=BC=3~(1/2)r,AB=2CD=2r,OA 杆以匀角速度ω绕 O 轴转动,CD 杆以匀角速度2ω绕 D轴转动。在图示位置,OA⊥AB,CD∥AB,BC 与 AB夹角为60°.试求该瞬时各杆的瞬时速度中心位置和角速度.     

各向异性平面变形及横观各向同性轴对称变形的塑性极限平衡方程

一、塑性条件 在各向同性情况下,普遍的塑性条件形式可写成(1.1)或1/2(σ_1-σ_3)sin2δ=f[1/2(σ_1+σ_3)-1/2(σ_1-σ_3)cos 2δ],(1.2)式中δ为滑移面与σ_1同的夹角,df/dσ_n=ctg2δ(图1)。由式(L2)易得     

EPSILON-ALGORITHM AND ETA-ALGORITHM OF GENERALIZED INVERSE FUNCTION-VALUED PAD APPROXIMANTS USING FOR SOLUTION OF INTEGRAL EQUATIONS

IntroductionLetf(x ,λ)beagivenpowerserieswithfunction_valuedcoefficients,i.e.,f(x ,λ) =c0 (x) +c1(x)λ+c2 (x)λ2 +… +cn(x)λn+… ,( 1 )wherecj(x)isarealorcomplexfunctionwithregardasxwhichisdefinedon (a ,b) .Supposethatf(x,λ)isanalyticasafunctionofλattheoriginλ=0 .ItisknownthatPad啨ａｐｐｒｏｘｉｍａｎｔｍｅｔｈｏｄｓａｒｅｕｓｅｄｆｏｒｏｂｔａｉｎｉｎｇｔｈｅｓｏｌｕｔｉｏｎｏｆａｗｋｗａｒｄｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎｓ,andespeciallythosewhichpossessageneratingfunctionoftheform ( …     

An investigation of electromagnetic wave propagation in plasma by shock tube

This paper presents the electromagnetic wave propagation characteristics in plasma and the attenuation coefficients of the microwave in terms of the parameters he, v, w, L, wb. The φ800 mm high temperature shock tube has been used to produce a uniform plasma. In order to get the attenuation of the electromagnetic wave through the plasma behind a shock wave, the microwave transmission has been used to measure the relative change of the wave power. The working frequency is f = (2-35)GHz (ω=2πf, wave length A =15cm-8mm). The electron density in the plasma is ne = (3&#215;10^10-1&#215;10^14) cm^-3. The collision frequency v = (1&#215;10^8-6&#215;10^10) Hz. The thickness of the plasma layer L = (2-80)cm. The electron circular frequency ωb=eBo/me, magnetic flux density B0 = (0-0.84)T. The experimental results show that when the plasma layer is thick (such as L/λ≥10), the correlation between the attenuation coefficients of the electromagnetic waves and the parameters ne,v,ω, L determined from the measurements are in good agreement with the theoretical predictions of electromagnetic wave propagations in the uniform infinite plasma. When the plasma layer is thin (such as when both L and A are of the same order), the theoretical results are only in a qualitative agreement with the experimental observations in the present parameter range, but the formula of the electromagnetic wave propagation theory in an uniform infinite plasma can not be used for quantitative computations of the correlation between the attenuation coefficients and the parameters ne,v,ω, L. In fact, if ω&lt;ωp, v^2&lt;&lt;ω^2, the power attenuations K of the electromagnetic waves obtained from the measurements in the thin-layer plasma are much smaller than those of the theoretical predictions. On the other hand, if ω&gt;ωp, v^2&lt;&lt;ω^2 (just v≈f), the measurements are much larger than the theoretical results. Also, we have measured the electromagnetic wave power attenuation value under the magnetic field and without a magnetic field. The result indicates that the value measured under the magnetic field shows a distinct improvement.     

On the existence of periodic solutions for nonlinear system with multiple delays

IntroductionInthispaper,westudyT_periodicsolutionsofthefollowingnonlinearsystemwithmultipledelays x(t) =f(t,x(t) ,x(t-τ1(t) ) ,… ,x(t -τm(t) ) ) ,(1 )wherex(t) ∈C(R ,R) ,fiscontinuous,f(t+T ,·) =f(t,·) ,τi(t) (i=1 ,2 ,… ,m)arecontinuousperiodicfunctionsofperiodT .AlemmaisintroducedfordiscussingtheexistenceofT_periodicsolutionofsystem (1 ) .LetXbeaBanachSpace ,considerthefollowingoperatorequation :Lx =λNx　　 (λ∈ [0 ,1 ] ) ,whereL :DomL∩X→Xisalinearoperator,λ∈ [0 ,1 ]isapa…     

关联噪声驱动非对称双稳系统的MFTP

考虑了由关联噪声驱动的非对称双稳系统的平均首次穿越时间(MFPT)问题,运用最速下降法求得平均首次穿越时间的表达式,并讨论了各个参数(包括p,q,r,λ)对两个方向的平均首次穿越时间的影响.结果表明:1)噪声关联时间λ对平均首次穿越时间T (x1→x2,λ,r)和T-(x2→x1,λ,r)的影响是不同的,T 随着λ的增加而增加,而T-却随着λ的增加而减小.2)平均首次穿越时间 T (x1→x2,λ,r)和T-(x2→x1,λ,r)都随加性噪声强度q的增加而减小;但乘性噪声P对T 和T-影响却是完全不同的,T-随着P的增加而减小,而T 随着P的增加曲线有一个极大值,即是一个共振峰.3)在不同的噪声关联强度的影响下,T /T-的曲线随着P的增加呈现不同的发展趋势,λ=0.1时,T /T-随着P的增加单调减小;而λ＝0.7时,T /T-随着P的增加先增加再减小,曲线出现了极值峰.     

The floquet theory for quasi-periodic linear systems

In this paper we establish the Floquet theory for the quasi-perio-dic systemwhere A(u_1,u_2,…,u_m)is an n×n periodic matrix function of u_1,u_2.…,u_mwith period 2π,and it is of C~τ,τ=(N 1)τ_0,τ_0=2(m 1).N=(1/2)n(n 1).Meanwhile,we define the characteristic exponential roots β_1,β_2,…,β_nof(0.1),and assume thatwhere K(ω),K(ω,β)>0.k_μ,j_v.are integers,all the integers k_1,k_2,…,k_m.are not zero,i~2=-1,Then there exists aquasi-periodic linear transformation,which carries(0.1)into a li-near system with constant coefficients.     

The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＬｅｍｍａｓＴｈｅｒｅａｒｅｍａｎｙｒｅｓｕｌｔｓａｂｏｕｔｅｘｉｓｔｅｎｃｅ (ｇｌｏｂａｌｏｒｌｏｃａｌ)ａｎｄａｓｙｍｐｔｏｔｉｃｂｅｈａｖｉｏｒｏｆｓｏｌｕｔｉｏｎｓｆｏｒｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ[1- 9].Ｂｙｔｈｅａｉｄｓｏｆｒｅｓｕｌｔｓ[2 ,3]ｏｆｅｑｕａｔｉｏｎ ｕ/ ｔ=Δｕ-λ｜ｕ｜γ- 1ｕｗｉｔｈｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｓ,ｐａｐｅｒ [4 ]ｓｔｕｄｉｅｄｔｈｅｐｒｏｂｌｅｍｏｆ ｕ/ ｔ=Δｕ-λ｜ｅβｔｕ｜γ- …     

低雷诺数流体力学介绍

、概述设物体在流体中运动,其尺度为 a,运动速度为 U,流体的密度为ρ,粘性系数为μ.μ具有质量/长度×时间的量纲(ML~(-1)T~(-1)).对于水,μ=10~(-2)泊(Poise).流体的运动粘性系数为ν,ν=μ/ρ,量纲为长度~2/时间(L~2T~(-1)).对于水,ν=10~(-2)厘米~2/秒=1 centistokes.     

A family of integrable systems of Liouville and lax representation,Darboux transformations for its constrained flows

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｗｅｌｌ＿ｋｎｏｗｎｂａｓｉｓｏｆｌｏｏｐａｌｇｅｂｒａ Ａ1ｉｓｔｈａｔｈ(ｎ) =λｎ 　 00 -λｎ ,　ｅ(ｎ) =0λｎ0 0 ,ｆ(ｎ) =0 0λｎ 0 ,[ｈ(ｍ) ,ｅ(ｎ) ] =2ｅ(ｍ+ｎ) ,[ｈ(ｍ) ,ｆ(ｎ) ] =　　 -2ｆ(ｍ+ｎ) ,[ｅ(ｍ) ,ｆ(ｎ) ] =ｈ(ｍ+ｎ) .( 1 )Ｍａｎｙｅｑｕａｔｉｏｎｈｉｅｒａｒｃｈｉｅｓｗｅｒｅｏｂｔａｉｎｅｄｂｙｅｓｔａｂｌｉｓｈｉｎｇｌｉｎｅａｒｉｓｏｓｐｅｃｔｒａｌｐｒｏｂｌｅｍｖｉａｔｈｅｕｓｅｏｆ( 1 ) ,ｓｕｃｈａｓＡＫＮＳｈｉｅｒａｒｃｈｙ ,ＫＮｈｉｅｒａ…     

Existence of the minimal positive solution of some nonlinear elliptic systems when the nonlinearity is the sum of a sublinear and a superlinear term

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｅｌｌｉｐｔｉｃｓｙｓｔｅｍ(1λ)　-Δｕ=ｆ(λ,ｘ,ｕ)-ｖ　　(ｉｎΩ),-Δｖ=δｕ-γｖ(ｉｎΩ),ｕ=ｖ=0(ｏｎΩ),ｗｈｅｒｅΩｉｓａｓｍｏｏｔｈｂｏｕｎｄｅｄｄｏｍａｉｎｉｎＲＮ(Ｎ≥2)ａｎｄλｉｓａｒｅａｌｐａｒａｍｅｔｅｒ.Ｔｈｅｓｏｌｕｔｉｏｎｓ(ｕ,ｖ)ｏｆｔｈｉｓｓｙｓｔｅｍｒｅｐｒｅｓｅｎｔｓｔｅａｄｙｓｔａｔｅｓｏｌｕｔｉｏｎｓｏｆｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｓｙｓｔｅｍｓｄｅｒｉｖｅｄｆｒｏｍｓｅｖｅｒａｌａｐ…     

Chaotic belt phenomena in nonlinear elastic beam

IntroductionThechaoticphenomenainsolidmechanicsfieldsbringmoreandmoreinterest.In 1 998,F .C .Moon[1]analyzedthechaoticbehaviorsofbeamsexperimentallyfirst.Thenhestudiedthedynamicsresponseoflinearelasticbeamsubjectedtransverseperiodicload .Thechaoticmotionsoflineardampingbeamshavebeenstudiedbymanyscholarsathomeandabroadinrecentyears[2 ,3].ThedynamicbehaviorsofnonlineardampingbeamssubjectedtotransverseloadP=δP0 (f+cosωt)sin(πx/l)arestudiedinthispaper.Thecriticconditionsthatchaosoccursinthes…     

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     

三维波浪型斜拉索横向受迫振荡数值研究

采用动网格技术针对低雷诺数(Re=100)下三维波浪型斜拉索在均匀来流中横向受迫振荡问题进行了数值模拟分析。引入直拉索做对比,探讨低雷诺数下不同波浪形状波长参数(λ/D_m为2、6)时三维波浪型斜拉索在特定的振荡振幅(A/D_m=0.2)及不同的振荡频率(fe/fs为0~2)下,其表面所受作用力、尾流流动结构和锁定区间振动响应的变化规律。研究结果表明:λ/D_m=2波浪型斜拉索类似于直拉索,在一定振荡频率范围内(fe/fs为0.9~1.22)同样会发生"锁定"现象,有一定的抑振效果,但没有明显的减阻效果;λ/D_m=6波浪型斜拉索在该区域仍是自然频率起主导作用,其脉动升力系数趋近于0.1,具有非常明显的减阻效果(减阻幅度可达25%),说明波浪型斜拉索三维表面形状的改变可直接影响其表面作用力、尾流流动模式转换及锁定等现象,特定形状的波浪型斜拉索能明显减阻且抑制其锁定区间的振动诱导。