动力系统前沿趋势研讨会，召集人：尤建功、龙以明、胡锡俊

会议日程

报告摘要

薛金鑫（清华大学）

The closing lemma and KAM normal form

 Abstract: TBA

刘会（武汉大学）

Closed orbits on Hamiltonian energy hypersurfaces and contact manifolds

Abstract: It’s one central problem for finding periodic orbits on Hamiltonian energy hyper‑ surfaces in Hamiltonian dynamics, which can be traced back to the work of A.M. Liapunov in 1892. With the development of symplectic geometry and contact topology in the past four decades, the periodic orbit problems are also concerned by geometers and topologist since it corresponds to closed Reeb orbit on contact manifold. In this lecture, I will firstly introduce this periodic orbit problem and review some classical results on this topic, at the same time several methods towards these problems will be mentioned. For the last part of this lecture, I will also report our recent progresses.

高婷（华中科技大学）

多尺度随机动力系统的有效动力学研究

Abstract: 多尺度随机动力系统由于能够在许多实际应用中描绘复杂现象，因此已广泛用于各种科学和工程问题。慢不变流形通常用来刻画系统的长时间慢尺度动力学性态， 故我们研究了一类数据驱动下，捕获快慢随机系统的慢不变流形及其约化系统的有效逼近方法。给定满足一些未知的慢‑快随机系统的短期观察数据，我们提出了一种新的算法，包括称为 Auto‑SDE 的神经网络，以学习不变的慢流形及其上的约化随机动力系统。我们的算法也在各种评估指标下的数值实验中被验证为准确、稳定和有效。

丛洪滋（大连理工大学）

The existence of full dimensional tori for Hamiltonian PDEs

Abstract: In this talk, we will discuss the existence of full dimensional tori for Hamiltonian PDEs.

许璐（吉林大学）

Degenerate nonlinear oscillator equations

Abstract: For a quasi‑periodically forced differential equation, response solutions are quasi‑ periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi‑ periodically forced oscillators. These solutions are well‑understood in quasi‑periodically per‑ turbed nonlinear oscillators either in the presence of large damping or in the non‑degenerate cases with small or free damping. This talk will present some recent results on the existence of responsive solutions in degenerate, quasi‑periodically forced nonlinear oscillators with small or free damping. The case with noise perturbation will also be discussed.

高美娜（上海第二工业大学）

Quasi‑periodic solutions around plane wave of high dimensional nonlinear Schrodinger equation

Abstract: In this paper, a degenerate KAM theorem with multiple normal frequencies is es‑ tablished under qualitative non‑degenerate conditions. As an application, quasi‑periodic so‑ lutions around plane wave are obtained for high dimensional nonlinear Schrodinger equa‑ tion with periodic boundary conditions.

段华贵（南开大学）

Index theorem and the multiplicity of periodic orbits

Abstract: In this talk, I will introduce two kinds of periodic orbit problems, i.e., closed geodesics on Finsler manifolds and closed orbits on hypersurfaces in R²ⁿ. Then I will simply explain how to deal with these problems by using the enhanced common index jump theorem recently es‑ tablished.

孙华清（东北大学）

On classification of singular matrix difference equations of mixed order

Abstract: This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl’s method by selecting a suitable quasi‑difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off‑diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.

周喆（中国科学院数学与系统科学研究院）

Almost periodic measures and rotation number of the Schroedinger operator with measure‑valued potentials

Abstract: 本报告中，我将汇报如何对一类几乎周期测度位势的薛定谔算子引入一个基本的动力学量——旋转数。如时间允许，再汇报一点后续问题。本报告基于和 David

Damanik、孟钢和章梅荣老师的工作。

吕克宁（四川大学）

Smooth Conjugate for Random Dynamical System

Abstract: TBA

王婧（南京理工大学）

拟周期驱动圆周同胚的分类相关问题研究

Abstract: 在本报告中，我们根据拟周期驱动圆周同胚纤维旋转数和驱动频率的关系，从有理相关和有理无关两个方面分别探讨了拟周期驱动圆周同胚不变曲线的存在性、锁模平台的稠密性，以及系统的线性化问题和线性化（可约性）的相关应用。本报告基于和黄文、Jaeger、Krikorian、王之任、周麒、尤建功老师的工作。

周青龙（浙江大学）

The positive fundamental group of Sp(2n)

Abstract: In this talk, we examine the homotopy classes of positive loops in Sp(2n). We show that two positive loops are homotopic if and only if they are homotopic through positive loops. This is a joint work with Jian Wang.

柳振鑫（大连理工大学）

Wasserstein convergence rates in the invariance principle for deterministic/sequential dynamical systems

Abstract: In this talk, we will discuss the convergence rate with respect to Wasserstein dis‑ tance in the invariance principle for dynamical systems with some hyperbolicity, where both deterministic case and sequential case are included. This is a joint work with Zhe Wang.

 韦屏远（北京大学）

Stochastic Hamiltonian Systems: Geometric Structure and Averaging Principle

Abstract: In this talk, we consider a Bismut class of stochastic Hamiltonian systems. We first show that such systems preserve characteristic leaves and structures. Based on these properties, we further investigate the effective behavior of a small transversal perturbation to a completely integrable stochastic Hamiltonian system with non‑Gaussian Lévy noise. We establish an averaging principle in the sense that the action component of solution converges to the solution of a stochastic differential equation when the scale parameter goes to zero.

刘磊（山东大学）

A symplectic dynamics approach to the spatial isosceles three‑body problem

Abstract: We study the spatial isosceles three‑body problem from the perspective of Symplec‑ tic Dynamics. For certain choices of mass ratio, angular momentum, and energy, the dynam‑ ics on the energy surface is equivalent to a Reeb flow on the tight three‑sphere. We find a Hopf link formed by the Euler orbit and a symmetric brake orbit, which spans an open book decomposition whose pages are annulus‑like global surfaces of section. In the case of large mass ratios, the Hopf link is non‑resonant, forcing the existence of infinitely many periodic orbits. The rotation number of the Euler orbit plays a fundamental role in the existence of periodic orbits and their symmetries. We explore such symmetries in the Hill region and show that the Euler orbit is negative hyperbolic for an open set of parameters while it can never be positive hyperbolic. Finally, we address convexity and determine for each parameter whether the energy surface is strictly convex, convex, or non‑convex. Dynamical consequences of this fact are then discussed.

程红玉（天津工业大学）

Global rigidity for ultra‑differentiable quasiperiodic cocycles

Abstract: We show that for a large class of ultra‑differentiable potential, and every irrational frequency, and for almost every energy E the corresponding quasiperiodic Schrödinger cocy‑ cle is either rotations reducible or has positive Lyapunov exponents. This partially answers the open problem by Fayad‑Krikorian. As spectral application, we prove the “Last’s inter‑ section spectrum conjecture”for Gervey potential, which answers an open problem raised by Jitomirskaya‑Marx.

况闻天（山东大学）

Periodic orbits of the Stark problem

Abstract: The Stark problem is a gravitational one body problem with an external force field. It is also known as the accelerated Kepler problem. We prove the existence of a sequence of symmetric brake orbit in Stark problem via shooting method. This is a joint work Ku‑Jung Hsu.

阮士贵（美国迈阿密大学）

Principal Spectral Theory and Asynchronous Exponential Growth for Age‑structured Models with Nonlocal Diffusion of Neumann Type

 Abstract: We study the principal spectral theory and asynchronous exponential growth for age‑structured models with nonlocal diffusion of Neumann type.  First, we provide two gen‑

eral sufficient conditions to guarantee existence of the principal eigenvalue of the age‑structured operator with nonlocal diffusion. Then we show that such conditions are also enough to en‑

sure that the semigroup generated by solutions of the age‑structured model with nonlocal diffusion exhibits asynchronous exponential growth. Compared with previous studies, we prove that the semigroup is essentially compact instead of eventually compact, where the latter is usually obtained by showing the compactness of solution trajectories. Next, follow‑ ing the technique developed in Vo (Math. Nach. 2022) we obtain some limit properties of the principal eigenvalue with respect to the diffusion rate and diffusion range. Finally, we estab‑ lish the strong maximum principle for the age‑structured operator with nonlocal diffusion. (Based on H. Kang & S. Ruan, Math. Ann. 2022).

欧昱伟（山东大学）

Linear Stability of Elliptic Relative Equilibrium in Spatial n‑body Problem via index theory

Abstract: It is well known that a planar central configuration of the n‑body problem gives rise to a solution where each particle moves on a Keplerian orbit with a common eccentricity e ∈ [0, 1), we call this solution an elliptic relative equilibrium (ERE for short). Since each particle of the ERE is always in the same plane, it’s natural to regard it as a planar n‑body problem. But in practical applications, it is more meaningful to consider the ERE as a spatial n‑body problem (i.e. each particle belongs to R³ ). In this talk, we give an expression of the spatial part and further get a rigorous analytical method to study the linear stability of the spatial part by the Maslov‑type index theory. As an application, we obtain the stability results of some classical ERE, including the elliptic Lagrangian solution, Euler solution and 1 + n‑gon solution

王楷植（上海交通大学）

弱 KAM 理论最新进展

Abstract: 我将首先介绍弱 KAM 理论产生的背景，发展脉络，及最初的经典结论。然后汇报我们在弱 KAM 理论方面的若干工作：时间周期哈密顿系统的弱 KAM, 切触哈密顿系统的弱 KAM, 以及弱 KAM 方法在相关问题中的应用。本报告基于报告人与多位学者合作的工作。

朱朝锋（南开大学）

A formula of local Maslov index and applications

Abstract: In this paper, we explicitly express the local Maslov index by a Maslov index in finite dimensional case without symplectic reduction. Then we calculate the Maslov index for the path of pairs of Lagrangian subspaces with continuously varying intersection to a fixed complemented Lagrangian subspace.  In particular,  we get the Maslov‑type index of  a given symplectic path in triangle form. We study the continuity of families of bounded linear relatioins and families of bounded linear operators acting on closed linear subspaces as technique preparations.

秦文新（苏州大学）

具有零拓扑熵的单调扭转映射的 Denjoy 极小集之唯一性

Abstract: 对于零拓扑熵的单调扭转映射，我们证明以无理数为旋转数的回复点集可以由一个圆周上的保向同胚来描述。因此，对任意无理旋转数，或者存在一个不变圆，或者存在唯一的 Denjoy 极小集。

张建路（中国科学院数学与系统科学研究院）

广义哈密顿‑雅可比方程的选择性原理

Abstract: Lions‑Papanicolaou‑Varadhan 在 1987 年引入了哈密顿‑雅可比方程的齐次化方法，用以得到系统稳态的 effective Hamiltonian。这一极值在动力学意义下等价于 Mather 的 alpha 函数。此外，L‑P‑V 提出利用 discount 消失可以等价地得到同一极值，并进一步提出了这一消失过程中粘性解收敛性的开放问题。我们将在广泛意义下，证明这一收敛性。

蔡傲（苏州大学）

混合随机拟周期斜积系统的 Lyapunov 指数

Abstract: 在本报告中，为了研究拟周期系统的随机扰动问题，我们将引入混合随机拟周期斜积系统的概念，介绍其 Lyapunov 指数的相关性质，包括其正性判别法则与连续性结果，并与纯随机与纯拟周期情形进行比对，展示其变化趋势。

余国巍（南开大学）

Hyperbolic motions in the n‑body and the restricted (n+1)‑body problems

Abstract: In the mathematical study of celestial mechanics, some of the most important re‑ sults are about final motions of the celestial bodies as time goes to infinity. The motion is called hyperbolic if the body goes to infinity with some definite asymptotic direction and no‑zero asymptotic velocity, as in the Kepler problem. In this talk, we survey some recent progress for hyperbolic motions in the n‑body problem and the restricted n+1 body problem.

于翔（西南财经大学）

On Periodic Orbits of the Planar N‑body Problem

Abstract:By introducing a novel coordinate system, we prove that there are abundant new periodic orbits near relative equilibria of the planar N‑body problem.

肖华峰（广州大学）

Periodic solutions of differential equations with distributed delay

Abstract: We studied periodic solutions of differential equations with distributed delay (DDDEs for short). Utilizing of Kaplan‑Yorke’s method, we transfered the search for periodic solu‑ tions of DDDEs into that of finding periodic solutions of ordinary differential systems (ODE for short). Making use of variational method, we obtains some sufficient conditions to guar‑ antee the existence and multiplicity of periodic solutions of DDDEs.

周茂林（南开大学）

Principal eigenvalue of second order elliptic and parabolic operators with large advection

Abstract: In this talk, we will discuss the recent progress on the limit problem of principal eigenvalue of second order operator in the last three years：1. the elliptic operator with degenerate advection; 2. the parabolic operator in 1 dimensional case; 3. breakthrough in higher dimensional cases.

常小军（东北师范大学）

Normalized solutions of mass supercritical NLS on non‑compact metric graphs with localized nonlinearities

Abstract: This talk is devoted to the existence of non‑trivial bound states of prescribed mass for the mass supercritical nonlinear Schrödinger equation on noncompact metric graphs with localized nonlinearities. The investigation is based upon a general variational principle which combines the monotonicity trick and a min‑max theorem with second order informa‑ tion, and upon the blow‑up analysis of bound states with prescribed mass and bounded Morse index.This talk is based on joint works with J. Borthwick (McGill University), L. Jeanjean (Uni‑ versity of Franche‑Comté) and N. Soave (Università degli Studi di Torino).

王宇辰（天津师范大学）

Doubly‑connected uniformly rotating vortex patches and their boundary regularity

Abstract: In this talk we consider rotating vortex patches (the V‑states) on the plane which are relative equilibrium of the two dimensional incompressible Euler equation. With a de‑ tailed analysis on the nonlinear structure of the functional, we obtain families of doubly‑ connected rotating vortex patches emanating from specific annuli by solving a highly degen‑ erate bifurcation problem，where the kernel is 2‑dim and the transversality condition fails. Moreover, we studied the singular structures of the limiting V‑states.