Instability of parametrically second- and third-subharmonic resonances governed by nonlinear Shrödinger equations with complex coefficients

A theoretical analysis of the parametric harmonic response of two resonant modes is made based on a cubic nonlinear system. The analysis based on the method of multiple scales. Two types of the modified nonlinear Schrödinger equations with complex coefficients are derived to govern the resonance wave. One of these equations contains the first derivatives in space for a complex-conjugate type as well as a linear complex-conjugate term that is valid in the second-harmonic resonance cases. The second parametric equation contains a complex-conjugate type which is valid at the third-subharmonic resonance case. Estimates of nonlinear coefficients are made. The resulting equations have an interesting in many dynamical and physical cases. Temporal modulational method is confirmed to discuss the stability behavior at both parametric second- and third-harmonic resonance cases. Furthermore, the Benjamin–Feir instability is discussed for the sideband perturbation. The instability behavior at the sharp resonance is examined and the existence of the instability is found.     

Rotating orbits of a parametrically-excited pendulum

The authors consider the dynamics of the harmonically excited parametric pendulum when it exhibits rotational orbits. Assuming no damping and small angle oscillations, this system can be simplified to the Mathieu equation in which stability is important in investigating the rotational behaviour. Analytical and numerical analysis techniques are employed to explore the dynamic responses to different parameters and initial conditions. Particularly, the parameter space, bifurcation diagram, basin of attraction and time history are used to explore the stability of the rotational orbits. A series of resonance tongues are distributed along the non-dimensionalied frequency axis in the parameter space plots. Different kinds of rotations, together with oscillations and chaos, are found to be located in regions within the resonance tongues.     

Cauchy problem for Mathieu’s equation at parametric resonance

Mathieu’s equation is solved by an asymptotic averaging method in the fourth approximation for the first to fourth resonance domains and in the third approximation for the zero resonance domain. The general periodic and aperiodic solutions on characteristic curves are found, and the general solution is obtained in instability domains and stability-domain areas adjacent to the characteristic curves. All the solutions are explicitly found in the form of functions of an argument without using the auxiliary parameter employed in Whittaker’s method. Simple formulas depending on two parameters of the equation are derived for the characteristic exponent in instability domains and for the frequency of slow oscillations in stability domains near the characteristic curves. The theory is developed by analyzing the resonances exhibited by Mathieu’s equation.     

A pendulum with an elliptic-type parametric excitation: Stability charts for a damped and undamped system

In this paper, a pendulum parametrically excited by the excitation which has the form of the Jacobi cn elliptic function is considered. Three cases related to the value of the elliptic parameter are distinguished: the case when it is smaller than zero, when it ranges between zero and unity, and when it is higher than unity. First, interpretations of the excitation with such elliptic parameter are given in terms of its period, higher harmonic content and the amplitude. These interpretations enable one to consider the elliptic-type excitation as a type of multi-cosine excitation whose frequency and amplitude are related mutually in a particular way. Stability charts are determined for damped and undamped systems. When the elliptic parameter is equal to zero, the governing equations considered transform to the well-known Mathieu equation. In all other cases, the governing equations considered can be seen as a new generalisation of the Mathieu equation. The influence of an arbitrary real elliptic parameter on the location and shape of the transition curves and instability tongues is investigated, illustrated and discussed in all three cases, which represent new and so far unknown results.     

初始缺陷和拉-弯耦合对于叠层板的振动、屈曲和非线性动力稳定性的影响

研究了复合材料叠层板的初始缺陷和拉伸＿弯曲耦合对于其振动、屈曲和非线性动力稳定性的影响·推导了控制方程·这是一个修正的非线性Ｍａｔｈｉｅｕ方程·进行了５种典型复合材料的数值计算,它们是玻璃环氧Ｓｃｏｔｃｈ＿１００２,芳纶环氧Ｋｅｖｌａｒ＿４９,硼环氧Ｂ４／５５０５,石墨环氧Ｔ３００／５２０８,和ＡＳ／３５０１·结果表明：由于初始缺陷耦合效应的存在,使叠层板对于进入参数共振更加敏感,并且其振幅大于无初始缺陷或者无耦合效应的叠层板·对于不同复合材料的叠层板,尤其是层数较少的板,耦合效应是不相同的·在板结构的屈曲和动力稳定性设计中,如果忽略了耦合效应的影响,其不安全性将超过１０％以上     

Mathieu equation with application to analysis of dynamic characteristics of resonant inertial sensors

In this paper, Mathieu equation is applied to analyze the dynamic characteristics of resonant inertial sensors. Unlike previous work, Mathieu equation is not just a differential equation and analyzes the stability of the transition curves, but become an important method in analyzing parametric resonant characteristics and approximate output of resonant inertial sensors. It is demonstrated that the mathematical model of resonant inertial sensors is described by Mathieu equation. The relevant Mathieu equation theory and dynamic characteristics analysis methods were proposed, which include both stability and dynamic linear output. Finally, theoretical and experimental analysis show that the correlation of the theoretical curve and the experimental result coincide so perfectly, which means proposed analysis methods for Mathieu equation could be used to analyze the dynamic output characteristic of resonant inertial sensors. The theoretical analyzing approach of Mathieu equation and experimental results of resonant inertial sensors are obtained, which provide an application area for Mathieu equation and a reference for the robust design for resonant inertial sensors.     

基于AP法的一类非线性机械动力系统的求解

用渐近摄动法分别一类机械系统的非线性运动控制方程1∶1、1∶2内共振主参数共振-1/2亚谐共振情况进行摄动分析,得到系统的平均方程.结果发现用渐近摄动法求得1∶2内共振的平均方程中会漏掉某些非线性项,而且内共振比值越大漏掉的项越多,由此可以看出渐近摄动法不适用于求解多模态之间内共振比值大的非线性动力学系统.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

Study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method

This paper deals with the study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method. The paper considers a micro-beam suspended between two conductive micro-plates, subjected to a same actuation voltage. The nonlinear governing differential equation of motion about static equilibrium position using calculus of variation theory and Taylor series expansion has been linearized and implementing a Galerkin based reduced order model a Mathieu type equation has been obtained. By improving variational iteration method combining with method of strained parameters transition curves, separating stable from unstable regions have been obtained. The results of variational iteration method, perturbation and direct numerical integration methods for some cases selected from different regions (stable and unstable regions) have been compared.     

A Finite-difference Solution of the Ginzburg–Landau Equation

Two finite-difference methods, which differ only in the way that they approximate the derivative boundary conditions, are developed for solving a particular form of the complex Ginzburg–Landau equation of superconductivity. The non-linear term in this equation is linearized in a way familiar to readers of Professor Mickens' work, and the numerical solution is obtained at each time step by solving a linear algebraic system. Consistency and stability are discussed and some numerical results are reported.     

Quasi-stationary solitons for Langmuir waves in plasmas

The multiple-scale perturbation analysis is used to study the perturbed nonlinear Schrödinger’s equation, that describes the Langmuir waves in plasmas. The perturbation terms include the non-local term due to nonlinear Landau damping. The WKB type ansatz is used to define the phase of the soliton that captures the corrections to the pulse where the standard soliton perturbation theory fails.     

Lévy-driven Volterra Equations in Space and Time

We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the Lévy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyze its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.     

参外激励下强非线性广义Van der Pol方程的亚谐共振解（英文）

In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitations by parametric transformation. Bifurcation response equation and transition sets of sub-harmonic resonance with strong nonlinearity of general Van der Pol equation with parametric and external excitation are worked out.Besides, transition sets and bifurcation graphs are drawn to help to analysis the problems theoretically. Conclusions show that the transition sets of general and nonlinear Van der Pol equation with parametric and external excitations are more complex than those of general and nonlinear Van der Pol equation only with parametric excitation, which is helpful for the qualitative and quantitative reference for engineering and science applications.     

Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation

We introduce a nonlinear perturbation technique to third order, to study the stability between two cylindrical inviscid fluids, subjected to an axial electric field. The study takes into account the relaxation of electrical charges at the interface between the two fluids. At first order, a linear dispersion relation is obtained. Analytical and numerical results for the overstability and incipient instability conditions are given. For perfect dielectric fluids, the electric field has a stabilizing influence, while for leaky dielectric fluids, the electric field can have either a stabilizing or a destabilizing influence depending on the conductivity and permittivity ratios of the two fluids. At higher order, a nonlinear dispersion relation (nonlinear Ginzburg–Landau equation) is derived, describing the evolution of wave packets of the problem. For leaky dielectric fluids near the marginal state, a nonlinear diffusion equation (nonlinear incipient instability) is obtained. For perfect dielectric fluids, two cubic nonlinear Schrödinger equations are obtained. One of these equations to determine a nonlinear cutoff electric field separating stable and unstable disturbance, whereas the other is used to analyze the stability of the system. It is found that the nonlinear stability criterion depending on the ratio of permittivity, Such effects can only be explained successfully in the nonlinear sense, as the linear analysis unsuccessful to inform about them.     

The influence of parametric excitation on floating bodies

In the present study, we investigate the possibility of stabilizing a floating body that is unstable without parametric excitation. We prescribe a non–stationary geometric constraint, that is, the centre of gravity is moved periodically in vertical direction. This periodic movement causes the parametric excitation described by a non–autonomous differential equation. By using the stability chart of the Mathieu equation, we can find sets of parameters where the stabilization of the floating body is realizable. This can help in the stabilization of a canoeist, but it can also cause stable ships to capsize. Numerical simulations and an experiment were also accomplished to confirm that the stability chart derived for the case of the floating body is realistic. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

广义特征值极小扰动问题的一类黎曼共轭梯度法

研究含参数$l$非方矩阵对广义特征值极小扰动问题所导出的一类复乘积流形约束矩阵最小二乘问题.与已有工作不同,本文直接针对复问题模型,结合复乘积流形的几何性质和欧式空间上的改进Fletcher-Reeves共轭梯度法,设计一类适用于问题模型的黎曼非线性共轭梯度求解算法,并给出全局收敛性分析.数值实验和数值比较表明该算法比参数$l=1$的已有算法收敛速度更快,与参数$l=n$的已有算法能得到相同精度的解.与部分其它流形优化相比与已有的黎曼Dai非线性共轭梯度法具有相当的迭代效率,与黎曼二阶算法相比单步迭代成本较低、总体迭代时间较少,与部分非流形优化算法相比在迭代效率上有明显优势.     

Stability analysis based on nonlinear inhomogeneous approximation

The asymptotic stability of zero solutions for essentially nonlinear systems of differential equations in triangular inhomogeneous approximation is studied. Conditions under which perturbations do not affect the asymptotic stability of the zero solution are determined by using the direct Lyapunov method. Stability criteria are stated in the form of inequalities between perturbation orders and the orders of homogeneity of functions involved in the nonlinear approximation system under consideration.     

轴向变速运动弦线的非线性振动的稳态响应及其稳定性

研究具有几何非线性的轴向运动弦线的稳态横向振动及其稳定性．轴向运动速度为常平均速度与小简谐涨落的叠加．应用Hamilton原理导出了描述弦线横向振动的非线性偏微分方程．直接应用于多尺度方法求解该方程．建立了避免出现长期项的可解性条件．得到了近倍频共振时非平凡稳态响应及其存在条件．给出数值例子说明了平均轴向速度、轴向速度涨落的幅值和频率的影响．应用Liapunov线性化稳定性理论,导出倍频参数共振时平凡解和非平凡解的不稳定条件．给出数值算例说明相关参数对不稳定条件的影响．     

Dark soliton dynamics under the complex Ginzburg–Landau equation

The dynamical properties of the complex Ginzburg–Landau equation are considered in the defocusing (normal dispersion) regime. It is found that under appropriate conditions stable evolution of dark solitons can occur. These conditions are derived using a newly developed perturbation theory that also reveals an important aspect of the dynamics: the formation of a shelf that accompanies the soliton and is an intricate part of its evolution. Further conditions to suppress this effect are also derived. These analytical predictions are found to be in excellent agreement with direct numerical simulations.     

Turing patterns induced by cross-diffusion in a 2D domain with strong Allee effect

In this work, we introduce a two-dimensional domain predator-prey model with strong Allee effect and investigate the Turing instability and the phenomena of the emergence of patterns. The occurrence of the Turing instability is ensured by the conditions that are procured by using the stability analysis of local equilibrium points. The amplitude equations (for supercritical case cubic Stuart–Landau equation and for subcritical quintic Stuart–Landau equation) are derived appropriate for each case by using the method of multiple time scale and show that the system supports patterns like squares, stripes, mixed-mode patterns, spots and hexagonal patterns. We obtain the asymptotic solutions to the model close to the onset instability based on the amplitude equations. Finally, numerically simulations tell how cross-diffusion plays an important role in the emergence of patterns.