威尼斯人

12月6日（星期二）

地点：  数学楼二楼学术报告厅

报告1：   题目:Exact Solutions in theInvariant Manifolds of the Generalized Integrable H/'{e}non-Heiles System andExact Traveling Wave Solutions of Klein-Gordon-Schr/"{o}dinger Equations

  报告人：         李继彬教授， 华侨大学

Abstract:Weconsider the exact explicit solutions for the famous generalizedH/'{e}non-Heiles (H-H) system. Corresponding to the three integrable cases, onthe basis of the investigation of the dynamical behavior and level curves ofthe planar dynamical systems, we find all possible explicit exact parametric representations of solutions in the invariant manifolds ofequilibrium points in the four dimensional phase space. These solutions containquasi-periodic solutions, homoclinic solutions, periodic solutions as well asblow-up solutions. Therefore, we answered the question: what are the flow inthe center manifolds and homoclinc manifolds of the generalizedH/'{e}non-Heiles (H-H) system. As an application of the above results,  weconsider the traveling wave solutions for the coupled $(n+1)-$dimensionalKlein--Gordon-Schr/"{o}dinger Equations with quadratic power nonlinearity.

报告2：题目：Lins-de Melo-Pugh 猜想和它在 n=4 的证明

报告人：  李承治教授， 北京大学

摘要： 1977年, C. Lins, W. de Melo 和 C. C. Pugh 提出著名猜想：n 次经典 Li/'enard 方程至多有 $[/frac{n-1}{2}]$ 个极限环，并证明了猜想对 n=3 成立。此猜想在 n 大于等于 4 时是否仍成立，成为困扰人们三十多年的一个难题。直到2007年，F. Dumotier，D.Panazzolo 和 R. Roussarie 利用奇异摄动的方法，证明了这个猜想对 n=7 以及 n>7 的奇数不成立;2011年 P. Maesschalck 和 F.Dumortier 再次利用奇异摄动的方法，证明了这个猜想在 n大于等于6 时不成立。2012年李承治和 J.Llibre 证明了这个猜想对 n=4 成立。此猜想在 n=5 时是否成立，至今仍无结论。

我们拟对此猜想的相关背景和它在 n=4 时的证明作一个简要的介绍。

报告3：题目：On the dynamics ofquasi-periodically perturbed homoclinic solutions

报告人： 吕克宁教授， 四川大学，美国杨伯翰大学

摘要：We study the complicated dynamics ofquasi-periodically perturbed ordinary differential equations with a homoclinicorbit to a dissipative saddle point. We show that there are four regions ofparameters in which the equations have respectively: (1) attractingquasi-periodic integral manifolds of Levinson type; (2) transition to chaos;(3) strange attractors; (4) homoclinic tangles. In the case of homoclinictangles, we not only obtain the results on horseshoes similar to the existingones, but also give a comprehensive geometric description of the structures oftangles.

报告4：   题目： 平面系统分支函数的光滑性

报告人： 韩茂安，  上海师范大学

报告5：  题目：Linearization of nonautonomousdifferential equations

报告人：   夏永辉教授，华侨大学

Abstract:This talk presents some recent advance on the linearization of differentialequations. We studied the global linearization of the nonautonomous system$/dot{x}=A(t)x+f(t,x,/theta)$ under parameter variation when the linear system$/dot{x}=A(t)x$ admits a nonuniform exponential dichotomy. Weaker conditionsare established for the existence of topological conjugacy between linear andnonlinear systems. We weaken the Lipshchizian requirement in theGrobman-Hartman type theorem [Theorem 7,Luis-JFA2007,pp334-335] to theH/"older continuity and estimate a lower upper bound of the H/"olderexponent to guarantee the $C^0$ linearization. Further, we discuss on theregularity of the conjugation in $x$, $t$ and the parameter $/theta$.

报告6： 题目：Ergodic  averages on level sets and observablemeasures.

报告人， 赵云， 苏州大学

报告7：题目：三维向量场生成的流的动力学

报告人， 杨大伟， 苏州大学

摘要：    三维向量场的不变集合呈现出了非常高的复杂性，例如Lorenz由常微系统确定的“蝴蝶吸引子”。这类吸引子先后被刻画为“几何Lorenz吸引子”以及“奇异双曲吸引子”。我们将介绍三维向量场的Palis猜测：任何一个三维向量场或者可以被具有同宿切的向量场逼近，或者可以被具有整体奇异双曲结构的向量场逼近。

报告8：题目：Tail entropy of  measures

报告人， 廖刚， 苏州大学

摘要: We study the relationship between the tail entropy andthe partial hyperbolicity of invariant measures.  An upper bound of thetail entropy is given in terms of the partial hyperbolic index.