On multi-parameter bifurcation in diffusion models

The problem of bifurcation of periodic orbits from equilibrium when several parameters are present is discussed. The theory is developed from the viewpoint of differential equations on function spaces using the center manifold theory and the method of averaging. Theoretical and numerical analysis of a reaction-diffusion model is included.     

一类不连续平面二次微分系统的极限环

本文考虑了一类不连续平面二次可积非Hamilton微分系统在二次扰动下的极限环个数问题.利用一阶平均法,我们得到了从该系统中心的周期环域至少可以分支出5个极限环的结论.该结果表明不连续二次微分系统比其相应光滑微分系统至少可以多分支出2个极限环.     

Nonlinear dynamics and bifurcations in external feedback control of microcantilevers in atomic force microscopy

We study nonlinear dynamics and bifurcations in external feedback control of vibrating microcantilevers in atomic force microscopy. The efficiency and validity of the control methodology for microcantilevers was demonstrated and abundant nonlinear phenomena were observed in a previous numerical study. Using the averaging method and center manifold theory, we analyze a degenerate bifurcation of codimension two to explain a key feature of the previous numerical results. Numerical examples are also given, in which theoretical results are compared with numerical computations for bifurcations and unstable manifolds.     

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

This paper concerns the bifurcation of limit cycles for a quartic system with an isochronous center. By using the averaging theory, it shows that under any small quintic homogeneous perturbations, at most 14 limit cycles birfucate from the period annulus of the considered system.     

Centers of quasi-homogeneous polynomial differential equations of degree three

We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order.     

Bifurcation of limit cycles from the global center of a class of integrable non-Hamilton systems

In this paper, we consider the bifurcation of limit cycles for system $\dot{x}=-y(x^2+a^2)^m,~\dot{y}=x(x^2+a^2)^m$ under perturbations of polynomials with degree n, where $a\neq0$, $m\in \mathbb{N}$. By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from periodic orbits of the center of the unperturbed system. Particularly, if $m=2, n=5$, the sharp bound is 5.     

求解一类非线性方程的改进的平均法

本文研究确定非线性方程一致有效近似解的平均法,得到B方法(Krylov-Bogoliubov method)与KBM方法(Krylov-Bogoliubov-Mitropolski method)的改进形式,通过两个例子与多尺度方法的比较,说明改进的平均法的有效性,从而拓广了平均法的应用范围。     

基于最小二乘近似的模型平均方法

本文提出基于最小二乘近似的模型平均方法.该方法可用于线性模型、广义线性模型和分位数回归等各种常用模型.特别地,经典的Mallows模型平均方法是该方法的特例.现存的模型平均文献中,渐近分布的证明一般需要局部误设定假设,所得的极限分布的形式也比较复杂.本文将在不使用局部误设定假设的情形下证明该方法的渐近正态性.另外,本文将该方法发展到维数发散的情形.数值结果显示了该方法的优势.     

Normalization and averaging on compact lie groups in nonlinear mechanics

We consider the method of normal forms, the Bogolyubov averaging method, and the method of asymptotic decomposition proposed by Yu. A. Mitropol’skii and the author of this paper. Under certain assumptions about group-theoretic properties of a system of zero approximation, the results obtained by the method of asymptotic decomposition coincide with the results obtained by the method of normal forms or the Bogolyubov averaging method. We develop a new algorithm of asymptotic decomposition by a part of the variables and its partial case — the algorithm of averaging on a compact Lie group. For the first time, it became possible to consider asymptotic expansions of solutions of differential equations on noncommutative compact groups.     

On the convergence of some Procrustean averaging algorithms

Euclidean “(size-and-)shape spaces” are spaces of configurations of points in R ^N modulo certain equivalences. In many applications one seeks to average a sample of shapes, or sizes-and-shapes, thought of as points in one of these spaces. This averaging is often done using algorithms based on generalized Procrustes analysis (GPA). These algorithms have been observed in practice to converge rapidly to the Procrustean mean (size-and-)shape, but proofs of convergence have been lacking. We use a general Riemannian averaging (RA) algorithm developed in [Groisser, D. (2004) “Newton's method, zeroes of vector fields, and the Riemannian center of mass”, Adv. Appl. Math. 33, pp. 95–135] to prove convergence of the GPA algorithms for a fairly large open set of initial conditions, and estimate the convergence rate. On size-and-shape spaces the Procrustean mean coincides with the Riemannian average, but not on shape spaces; in the latter context we compare the GPA and RA algorithms and bound the distance between the averages to which they converge.     

正规形的复内积平均法

提出了求解非线性动力系统正规形的复内积平均法,与传统的平均法相比,其理论分析过程具有更加简单的形式,易于编程实现,可求到任意阶近似,并统一了自治系统与非自治系统的求解过程,最后求解一个例题来验证了该方法.     

QUASI-PERIODIC SOLUTIONS ANDSUB-HARMONIC BIFURCATION OF DUFFING'S EQUATIONS WITH QUASI-PERIODIC PERTURBATION

1.IntroductionInthispaperweconsidertheDuffing'sequationofthefollowingtype:whereg,fi,fZaresmallparameterswithorderO(E),6isasmallparameterwithorderO(E')'coscallac=afl ac,n=3,5,05e<<1.alandwZarenotrationalrelated,i.e.,f(t)=f,coswit jZcoswZtisaquasi-periodicfunction.Duffing'sequationwithoneexternalforcingtermiseXtensivelystudied,forexample,see[1--7].Duffing'sequationwithtwoperiodicexternalforcingtermsisalsoanalyzed,see[7-10].In[11,121theauthorsstudiedthedynamicsofaweaklynonlinearsystemsubjecte…     

Asymptotic behaviour of solutions of the difference Schrödinger equation

We use the method of averaging and the discrete analogue of Levinson's theorem to construct the asymptotics for solutions of the difference Schrödinger equation. Moreover, we present the general form for the averaging change of variable.     

Multi-instability of gap solitons and dynamics of nonlinear excitations in the array of optical fibers

The nonlinear localized excitations and gap multi-instability phenomena in the array of optical fibers are investigated analytically by means of the method of multiple scales combined with the averaging method. The perturbative analysis in nonlinear discrete systems leads to a system of four coupled nonlinear oscillated equations without second members. The method of multiple scales is used to show the different possibilities of instability. The averaging method is considered to show how the angular frequency received the perturbation with the evolution of time.     

The generalizedp-Centdian on network

Since the median approach is based on averaging, it often provides solutions where remote and low-population density areas are discriminated against in terms of accessibility to public facilities, as compared with centrally situated and high-population density areas. On the other hand, locating a facility at the centre may cause a large increase in total distance, thus generating a substantial loss in spacial efficiency. This has led to a search for some compromise solution concept. Halpern has introduced the λ-Centdian as a parametric solution concept based on the bicriteria center median model. He has modeled the corresponding trade-offs with a convex combination of the unweighted center and weighted median objectives. In this paper we study the λ-Centdian like a convex combination of the weighted center and weighted median objectives we called it generalized Centdian problem. This research has been partially supported by Gobierno de Canarias through projects PI1999/116.     

Multi-criteria decision-making method based on normal intuitionistic fuzzy-induced generalized aggregation operator

Normal intuitionistic fuzzy numbers (NIFNs), which use normal fuzzy numbers to express their membership and non-membership functions, can reflect the evaluation information exactly in different dimensions. In this paper, we are committed to apply NIFNs to multi-criteria decision-making (MCDM) problems, and meanwhile some new aggregation operators are proposed, including normal intuitionistic fuzzy weighted arithmetic averaging operator, normal intuitionistic fuzzy weighted geometric averaging operator, normal intuitionistic fuzzy-induced ordered weighted averaging operator, normal intuitionistic fuzzy-induced ordered weighted geometric averaging operator and normal intuitionistic fuzzy-induced generalized ordered weighted averaging operator (NIFIGOWA). Based on the NIFIGOWA operator, an approach is introduced to solve MCDM problems where the criteria values are NIFNs and the criteria weight information is fixed. Finally, the proposed method is compared to the existing methods by virtue of a numerical example to verify its feasibility and rationality.     

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

During the last three decades, the averaging method has been successfully applied to a wide range of problems involving differential inclusions, simplifying the study of the systems under consideration. In this work, the main development trends and methods in the application of the averaging method to the study of stability and optimality of solutions to differential inclusions are surveyed. A detailed list of references is given and some examples of applications are presented.     

LIMIT CYCLES OF QUARTIC AND QUINTIC POLYNOMIAL DIFFERENTIAL SYSTEMS VIA AVERAGING THEORY

We study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging theory. More precisely,we prove that the perturbations of the period annulus of the center located at the origin of a cubic polynomial differential system,by arbitrary quartic and quintic polynomial differential systems,there respectively exist at least 8 and 9 limit cycles bifurcating from the periodic orbits of the period annu...     

Quasi-periodic solutions and sub-harmonic bifurcation of Duffing’s equations with quasi-periodic perturbation

The quasi-periodic perturbation for the Duffing’s equation with two external forcing terms has been discussed. The second order averaging method and sub-harmonic Melnikov’s method through the medium of the averaging mrthod have been applied to detect the existence of quasiperiodic solutions and sub-harmonic bifurcation for the system. Sub-harmonic bifurcation curves are given by using numerical computation for sub-harmonic Melnikov’s function.     

基于对偶犹豫模糊语言变量的多属性决策方法

首先定义了对偶犹豫模糊语言变量,然后给出其运算规则、得分值函数、精确值函数、比较规则以及对偶犹豫模糊语言变量的加权算术平均算子、有序加权算术平均算子和混合平均算子。针对属性值为对偶犹豫模糊语言变量的多属性决策问题,提出了一种基于对偶犹豫模糊语言变量集结算子的多属性决策方法。最后,结合国家电网公司合作单位选择问题,验证了该方法的有效性和可行性。