具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔

研究了具有时滞耦合的n个van derPol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2:5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2:5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.     

具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔

研究了具有时滞耦合的n个van der Pol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2∶5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2∶5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.     

一类非线性振动系统的渐近解

本文提出分析一类强非线性振子的一种渐近方法,导出了用振幅和相位表示的二阶近似解,给出了振幅和相位所满足的方程,借此可以确定极限环的振幅和性态。作为实例,还研究了修正的van der Pol振子,给出了相应的极限环的二阶近似解析解,与数值解的比较表明,两者非常相符。     

集体坐标和高能集团散射理论

本文把“价”和“海”的概念引入多粒子系统,并利用Glauber散射理论把一个复杂的多粒子系统的多次散射振幅,分为价-价、价-海、海-海散射振幅。为了进行应用性研究,文中对每一种散射振幅采用一种有效的近似方法处理。例如利用独立粒子模型近似处理价-价散射振幅,利用A.Bohr的集体模型近似处理海-海散射振幅,以及利用光学模型近似处理价-海散射振幅等等。这就给出了物理图象清楚,并便于实际计算的多粒子系统的高能散射S矩阵元。     

一类Z_2对称五次微分系统的中心条件和极限环分支

本文研究了一类Z2对称五次微分系统的中心条件和小振幅极限环分支.通过前6阶焦点量的计算,获得了原点为中心的充要条件,并证明系统从原点分支出的小振幅极限环的个数至多为6.最后通过构造后继函数,给出系统具有6个围绕原点的小振幅极限环的实例.     

一类旋转磁对流系统振幅方程的双波前解的存在性和稳定性分析

研究了旋转磁对流系统的振幅方程的一类稳态解的存在性和稳定性问题,得出了振幅方程的双波前解的存在性定理及相容性条件,通过变分特征化方法证明了方程解的不稳定性.     

单自由度体系有阻尼自由振动振幅包络线的性质研究与数学证明

针对单自由度体系有阻尼自由振动,探究了振幅包络线与振动位移一时间曲线(以下简称振动曲线)的交点个数、振幅包络线与振动曲线交点位置以及振幅包络线值与结构阻尼比的关系三方面的问题.通过理论推导,给出了一条振幅包络线在单周期内与振动曲线只有一个切点且切点位于振动曲线峰值点稍靠右侧处的证明过程.通过改变相关参数的取值,发现了在某些初始条件下在振动初始阶段振幅包络线的绝对值不与结构的阻尼比呈负相关的有趣现象,并给出了直观的图像说明与理论证明.     

具有两个驰豫时间的热弹性立方晶体材料中平面波的传播

研究具有两个驰豫时间的、两个不同弹性和热性质的、广义传热立方晶体固体半空间的有缺陷结合面上,热弹性平面波的反射和折射问题.具有两个驰豫时间的广义热弹性理论,是1972年由Green和Lindsay提出并应用于问题的研究.对有缺陷边界,给出了反射系数和折射系数(即反射波和折射波振幅与入射波振幅之比)的表达式,并推演了法向刚性边界、横向刚性边界、接触传热边界、滑动边界和结合面边界时的表达式.给出了在不同边界条件及出射角时,不同的反射波和折射波的振幅比,在不同的入射波时的变化图.发现反射波和折射波振幅比受到介质刚性和热性质的影响.     

时间和振幅受限的最小能量控制问题

本文用初等的分析和代数方法 ,对于固定的控制时间 T,得到了线性系统控制振幅受限的最小能量控制所满足的一个充分条件 .举例说明了利用本文所得的结果 ,可以求得有限维线性控制系统控制振幅受限的最小能量控制函数的具体表达式 .     

平面P波在弹性介质和非饱和多孔弹性介质分界面上的传播

使用线性粘滞的多孔弹性介质模型,解决在弹性介质和非饱和多孔弹性介质分界面上平面P波的反射与透射问题,这里的非饱和多孔介质中固体骨架被两种相互耦合的流体（液体和气体）所充满．通过势函数的方法得到了振幅反射系数与振幅透射系数．然后推导得到入射波与反射波、透射波之间能量转换情况．研究发现:用振幅比和能量比所表示的反射系数与透射系数是与入射角度、饱和度、入射频率以及上下层介质的弹性常数有关的方程式．数值计算通过图形的形式表达出来,而且入射角度、频率及饱和度对振幅和能量的反射与透射系数的影响分别进行了讨论．证明了在整个波的传播过程中分界处并没有发生能量的耗散．     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Autoresonance excitation of a soliton of the nonlinear Schrödinger equation

The action of an external parametric perturbation with slowly changing frequency on a soliton of the nonlinear Schrödinger equation is studied. Equations for the time evolution of the parameters of the perturbed soliton are derived. Conditions for the soliton phase locking are found, which relate the rate of change of the perturbation frequency, its amplitude, the wave number, and the phase to the initial data of the soliton. The cases when the initial amplitude of the soliton is small and when the amplitude of the soliton is of the order of unity are considered.     

Autoresonance excitation of a breather in weak ferromagnetics

A mathematical model of the magnetodynamics of a weak ferromagnetic in an external magnetic field with variable frequency is studied. Conditions for the appearance of autoresonance are found under which the amplitude of the local magnetic inhomogeneities is considerably increased. Mathematically, the problem is to analyze the solutions to the sine-Gordon equation subject to a specific nonautonomous perturbation. For the perturbed equation, asymptotic solutions in the form of a breather with a slowly varying amplitude and a phase shift are constructed. The solutions whose amplitude increases with time from small values to quantities of order one are associated with the autoresonance phenomenon.     

Steady Gravity-Capillary Waves on Deep Water—II. Numerical Results for Finite Amplitude

Finite amplitude capillary-gravity waves of permanent form on deep water are studied numerically. Bifurcation and limit lines are calculated. Pure and combination waves are continued to maximum amplitude. It is found that the height is limited in all cases by the surface enclosing one or more bubbles.     

On the existence of optimum large amplitude hydrofoil propulsion

The possibility of optimum thrust-production by means of periodic motions of a flexible wing is discussed. The disturbance velocities are assumed to be small so that a linearized theory is valid. It is proved that when adequate constraints are put upon the amplitude of the path of the wing and on the amplitude of its slopes and curvature, an optimum path exists.     

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam are investigated in this paper. The beam is moving with a time-dependent velocity, namely a harmonically varied velocity about the mean velocity. The partial differential equation is discretized into nonlinear ordinary differential equations via the method of Galerkin truncation and then the steady-state response is obtained using the method of multiple scales, an approximate analytical method. The tuning equations are obtained by eliminating secular terms and the amplitude of the vibration is derived from the tuning equations expressed in polar form, and two bifurcation points are obtained as well. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. Eventually, the effects of various parameters such as the thickness of core layer, mean velocity, initial tension, and the amplitude of axially moving velocity on amplitude–frequency response curves and unstable regions are investigated.     

均布载荷作用下夹层圆板的非线性振动

给出了均布载荷作用下夹层圆板的大幅度振动方程.按假设的时间模态导出了该问题的非线性耦合的代数和微分特征方程组,并利用修正迭代法求出了该方程组的近似解析解,得到了周边固定夹层圆板振动的幅频-载荷特征关系.讨论了载荷对非线性振动性态的影响.