Oscillatory property of higher order nonlinear difference equations

In this paper, we study the oscillatory and asymptotic behaviour of solutions of higher order nonlinear difference equations of the form

We obtain some necessary and sufficient conditions for all bounded solutions of (*) to be oscillatory and for (*) to have a nonoscillatory solution of a special form.     

On a fallacious conjecture about the stabilizability properties of solutions of the Riccati difference equation

In [1], among other results, some conjectures concerning the monotonicity and stabilizing properties of solutions of the difference Riccati equation were proven to be fallacious by means of suitable counterexamples. Moreover, a ‘possibly fallacious conjecture’ was formulated for which no counterexample had been found. In this letter, such a counterexample is provided together with an interpretation of the somewhat counterintuitive behaviour of the Riccati equation.     

Solitary wave solutions for a higher order nonlinear Schrödinger equation

We consider a higher order nonlinear Schrödinger equation with third- and fourth-order dispersions, cubic–quintic nonlinearities, self steepening, and self-frequency shift effects. This model governs the propagation of femtosecond light pulses in optical fibers. In this paper, we investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the considered model. The derived analytical dark and bright wave solutions are expressed in terms of the model coefficients. These exact solutions are useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a higher-order nonlinear and dispersive Schrödinger system.     

Stability radii of higher order positive difference systems

In this paper we study stability radii of positive polynomial matrices under affine perturbations of the coefficient matrices. It is shown that the real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by some examples.     

Soliton perturbation theory for a higher order Hirota equation

Solitary wave evolution for a higher order Hirota equation is examined. For the higher order Hirota equation resonance between the solitary waves and linear radiation causes radiation loss. Soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and compared to numerical solutions. An excellent comparison between numerical and theoretical solutions is obtained for both right- and left-moving waves. Also, a two-parameter family of higher order asymptotic embedded solitons is identified.     

Non-oscillation of solutions of difference equations of third order

In this paper, sufficient conditions in terms of coefficient functions are obtained for non-oscillation of all solutions of a class of linear homogeneous third order difference equations of the form

     

微分方程在复杂网络社团发现中的研究

理解复杂网络的关键在于迅速精确地发现网络中的社团结构。基于图理论的谱聚类算法是一种有效并全局收敛的优秀社团发现算法,其计算量集中于特征值和特征向量的计算。结合常系数线性常微分方程的解与系数矩阵特征值的关系,提出了基于微分方程的谱聚类社团发现算法(AMCF和LMCF);这两种算法避免了矩阵的特征值和特征向量的复杂计算过程,为社团发现算法提供了新的思路。理论分析和实验验证了算法的有效性。     

利用JAVA小程序实现求解差分方程的动态演示

主要研究了如何使用迭代法求解阶差分方程，利用Java小程序内嵌网页，实现了多种激励下的输出序列(包括零输入响应、零状态响应、全响应)波形的动态绘制。界面交互性强，动态性好，丰富了差分方程的教学，对信号与系统课程的网上教学作了很好的尝试。     

具有动态边界阻尼的波方程的降阶型差分半离散化的一致指数稳定性

由于在最优控制的数值计算和可观性的反问题中,离散系统关于离散化参数的一致指数稳定性具有重要作用,因而一致指数稳定性得到广泛而深入的研究.众所周知,对指数稳定的波方程的空间变量用经典的有限差分或有限元离散化后,离散格式产生高频病态伪特征模,致使离散系统不是一致指数稳定的.为了恢复一致指数稳定性,学者们引入了添加数值粘性项法和滤波法等方法.然而对于具有动态边界条件的波方程的一致指数稳定性问题研究的较少.本文用降阶型差分方法对此问题进行研究,先对具有动态边界条件的波方程进行降阶处理,然后利用有限差分对其进行空间半离散化,不用再对其进行任何处理,引入合适的李雅普诺夫函数即可验证离散系统是一致指数稳定的.     

Monotonicity and stabilizability- properties of solutions of the Riccati difference equation: Propositions, lemmas, theorems, fallacious conjectures and counterexamples

The problem considered is that of selecting an initial covariance matrix for the Kalman filter to ensure that the closed-loop filter at every subsequent time instant is exponentially asymptotically stable as a time-invariant filter. Sufficient conditions are derived based on monotonicity properties of the solution of the Riccati difference equation. The results have application in observer design, and the cases of filtering for nonstabilizable systems and systems with singular system matrices are included.     

一类两种群竞争趋化模型解的有界性

研究了一类完全抛物型的含两竞争种群和一趋化物的Keller-Segel模型的非负解。在一些适当条件下,对充分光滑的初始条件,利用Moser型迭代可证得该模型存在唯一整体古典解,且有界。     

On stability of a class of positive linear functional difference equations

We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae. Pham Huu Anh Ngoc and Toshiki Naito are supported by the Japan Society for Promotion of Science (JSPS) ID No. P 05049.     

New exact solutions for the KdV equation with higher order nonlinearity by using the variational method

The Korteweg-de Vries (KdV) equation with higher order nonlinearity models the wave propagation in one-dimensional nonlinear lattice. A higher-order extension of the familiar KdV equation is produced for internal solitary waves in a density and current stratified shear flow with a free surface. The variational approximation method is applied to obtain the solutions for the well-known KdV equation. Explicit solutions are presented and compared with the exact solutions. Very good agreement is achieved, demonstrating the high efficiency of variational approximation method. The existence of a Lagrangian and the invariant variational principle for the higher order KdV equation are discussed. The simplest version of the variational approximation, based on trial functions with two free parameters is demonstrated. The jost functions by quadratic, cubic and fourth order polynomials are approximated. Also, we choose the trial jost functions in the form of exponential and sinh solutions. All solutions are exact and stable, and have applications in physics.     

On positive definite solutions to the algebraic Riccati equation

Complete necessary and sufficient conditions for the existence of a positive definite solution to the algebraic Riccati equation are given. It is also shown that when a positive definite solution exists, it is either unique, or else there are uncountably many such solutions.     

基于特殊差分方程的安全可验证门限秘密共享

在秘密共享方案的研究中,一般都进行Shamir（n,t）门限秘密共享方案的研究,该方案是基于多项式插值的门限方案。研究了基于特殊差分方程的门限秘密共享方案,同时,考虑了此类门限方案的安全性,最后基于特殊差分方程给出了安全可验证的门限秘密共享方案。可以得出,给出的门限秘密共享方案的信息率为1/2,且对于防欺诈是无条件安全的。     

Note on nonstability of the linear functional equation of higher order

We provide a complete solution of the problem of Hyers-Ulam stability for a large class of higher order linear functional equations in single variable, with constant coefficients. We obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characteristic equation is of module 1. Our results are related to the notions of shadowing (in dynamical systems and computer science) and controlled chaos. They also correspond to some earlier results on approximate solutions of functional equations in single variable.     

The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation

We investigate the periodic nature of the positive solutions of the fuzzy difference equation , where k, m are positive integers, A₀, A₁, are positive fuzzy numbers and the initial values x_i, i = −d, −d + 1, … , −1, d = max{k, m}, are positive fuzzy numbers. In addition, we give conditions so that the solutions of this equation are unbounded.