2017年动力系统及应用研讨会
主办单位： 大连理工大学数学科学学院
会议日期： 2017年5月11日14日
会议地点：大连星海高尔夫酒店
校内联系人：柳振鑫
Workshop Program

May 12 


Time 
Speaker 
Talk Title 


8:108:30 
Opening (Yufeng Lu), Taking photos 


Chair: Yong Li 


8:309:00 
Zeng Lian 
Periodic structure of quasiperiodic system 


9:009:30 
Wenmeng Zhang 
Differentiability of the conjugacy in the HartmanGrobman Theorem 


9:3010:00 
Jun Shen 
WongZakai approximations and center manifolds 


10:0010:20 
Tea break 


Chair: Fuzhong Cong 


10:2010:50 
Lijin Wang 
Numerical methods for stochastic Hamiltonian systems 


10:5011:20 
Yuanhong Wei 
Multiplicity of solutions for nonlocal elliptic equations driven by the fractional Laplacian 


11:2011:50 
Xu Wang 
Construction and convergence of an ergodic and conformal multisymplectic approximation for a damped stochastic NLS equation 


12:00 Lunch 


Chair: Jialin Hong 


14:0014:30 
David Cheban 
Levitan almost periodic solutions for stochastic differential equations 


14:3015:00 
Xiaohu Wang 
Asymptotic behavior of nonautonomous random reactiondiffusion equations on unbounded domains 


15:0015:30 
Jian Zu 
Optimal control methods for stability criteria on Hill's equations 


15:3015:50 
Tea break 


Chair: Shaoyun Shi 


15:5016:20 
Xingwu Chen 
Restricted independence in displacement function for estimating cyclicity 


16:2016:50 
Shuqing Liang 
The rate of decay of stable periodic solutions for Duffing equation with L^pconditions 


16:5017:20 
Liying Zhang 
Near preservation of quadratic invariants via explicit stochastic Runge–Kutta methods 


17:2017:50 
Jianbo Cui 
Strong Convergence Rate of Splitting Schemes for Stochastic Nonlinear Schrodinger Equations 


18:00 Dinner 

May 13 
Time 
Speaker 
Talk Title 
Chair: Kening Lu 
8:008:30 
Shuguan Ji 
Dynamical behavior of solutions for the wave equation in inhomogeneous media 
8:309:00 
Yixian Gao 
Analysis of transient acousticelastic interaction in an unbounded structure 
9:009:30 
Peng Gao 
Strong averaging principle for stochastic KleinGordon equation with a fast oscillation 
9:3010:00 
Lijun Miao 
Stochastic symplectic and multisymplectic methods for stochastic NLS equation with quadratic potential 
10:0010:20 
Tea break 
Chair: Shaoli Wang 
10:2010:50 
Xiaoyue Li 
Permanence and ergodicity of stochastic GilpinAyala population model 
10:5011:20 
Rui Gao 
Decay of correlations for Fibonacci unimodal interval maps 
11:2011:50 
Zhihui Liu 
Approximating stochastic partial differential equations with additive fractional noises 
12:00 Lunch 
Chair: Wen Huang 
14:0014:30 
Ji Li 
Invariant manifold and foliation for random dynamical system and application 
14:3015:00 
Feng Chen 
Almost automorphic solutions of mean field stochastic differential equations 
15:0015:30 
Liying Sun 
High order conformal symplectic and ergodic schemes for stochastic Langevin equation via generating functions 
15:3015:50 
Tea break 
Chair: Fengqi Yi 
15:5016:20 
Guanggan Chen 
Approximating dynamics of a singularly perturbed stochastic wave equation with a random dynamical boundary condition 
16:5016:50 
Libo Wang 
Minkowski空间中几类具奇性的平均曲率方程径向解的存在性 
16:5017:20 
Liu Liu 
Iterative roots of piecewise monotone functions 
17:2017:50 
Chuying Huang 
Symplectic RungeKutta methods for Hamiltonian systems driven by rough paths 
18:00 Dinner 









Abstracts of Workshop Talks
Levitan almost periodic solutions for stochastic differential equations
David Cheban
cheban@usm.md
State University of Moldova
Abstract: The talk is dedicated to studying the problem of Levitan almost periodicity of solutions for semilinear stochastic equation
dx(t)=(Ax(t)+f(t,x(t)))dt +g(t,x(t))dW(t)
with exponentially stable linear operator $A$ and Levitan almost periodic in time coefficients $f$ and $g$. We prove that if the functions $f$ and $g$ are appropriately ``small", then the above equation admits at least one Levitan almost periodic solution.
Almost automorphic solutions of mean field stochastic differential equations
Feng Chen（陈锋）
chenf101@nenu.edu.cn
Northeast Normal University
Abstract: In this talk, we consider a class of mean field stochastic differential equations. The existence and uniqueness of almost automorphic solutions in distribution of mean field stochastic differential equations are established provided the coefficients of equations satisfy some suitable conditions. The result is applied to study case of stochastic heat equations with mean field.
Approximating dynamics of a singularly perturbed stochastic wave equation with
a random dynamical boundary condition
Guanggan Chen（陈光淦）
chenguanggan@hotmail.com
Sichuan Normal University
Abstract: This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting is used to establish the approximating equation of the system for a sufficiently small singular perturbation parameter. The approximating equation is a stochastic parabolic equation when the power exponent of the singular perturbation parameter is in [1/2, 1) but is a deterministic wave equation when the power exponent is in (1,+∞). Moreover, if the power exponent of a singular perturbation parameter is bigger than or equal to 1/2, the same limiting equation of the system is derived in the sense of distribution, as the perturbation parameter tends to zero. This limiting equation is a deterministic parabolic equation. This work is jointed with Professor Jinqiao Duan and Professor Jian Zhang.
Restricted independence in displacement function for estimating cyclicity
Xingwu Chen（陈兴武）
xingwu.chen@hotmail.com
Sichuan University
Abstract: Since the independence of focal values is a sufficient condition to give a number of limit cycles arising from a centerfocus equilibrium, we consider a restricted independence to a parametric curve, which gives a method not only to increase the lower bound for the cyclicity of the centerfocus equilibrium but also to be available when those focal values are not independent.
Strong convergence rate of splitting schemes for stochastic nonlinear
Schrödinger equations
Jianbo Cui（崔建波）
jianbocui@lsec.cc.ac.cn
Chinese Academy of Sciences
Abstract: In this paper, we show that solutions of stochastic nonlinear Schrödinger (NLS) equations can be approximated by solutions of coupled splitting systems. Based on these systems, we propose a new kind of fully discrete splitting schemes which possess algebraic strong convergence rates for stochastic NLS equations. In particular, under very mild conditions, we derive the optimal strong convergence rate $\OOO(N^{2}+\tau^\frac12)$ of the spectral splitting CrankNicolson scheme, where $N$ and $\tau$ denote the dimension of the approximate space and the time step size, respectively. This is a joint work with Jialin Hong, Zhihui Liu, and Weien Zhou.
Strong averaging principle for stochastic KleinGordon equation
with a fast oscillation
Peng Gao（高鹏）
gaopengjilindaxue@126.com
Northeast Normal University
Abstract: This paper investigates an averaging principle for stochastic KleinGordon equation with a fast oscillation arising as the solution of a stochastic reactiondiffusion equation evolving with respect to the fast time. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different timescales. To be more precise, the wellposedness of mild solutions of the stochastic hyperbolicparabolic equations is firstly established by applying the fixed point theorem and the cutoff technique. Then, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic KleinGordon equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation.
Decay of correlations for Fibonacci unimodal interval maps
Rui Gao（高睿）
gaoruimath@scu.edu.cn
Sichuan University
Abstract: In this talk, we consider a class of (generalized) Fibonacci unimodal maps whose closest return times $\{s_n\}$ satisfy that $s_n=s_{n1}+\kappa s_{n2}$ for some $\kappa\ge 1$. We show that such a unimodal map admits a unique absolutely continuous invariant probability with stretched exponential decay of correlations precisely if its critical order lies in $(1,\kappa+1)$. This is a joint work with Weixiao Shen.
Analysis of transient acousticelastic interaction in an unbounded structure
Yixian Gao（高忆先）
gaoyx643@nenu.edu.cn
Northeast Normal University
Abstract: Consider the wave propagation in a twolayered medium consisting of a homogeneous compressible air or fluid on top of a homogeneous isotropic elastic solid. The interface between the two layers is assumed to be an unbounded rough surface. This paper concerns the timedomain analysis of such an acousticelastic interaction problem in an unbounded structure in three dimensions. Using an exact transparent boundary condition and suitable interface conditions, we study an initial boundary value problem for the coupling of the Helmholtz equation and the Navier equation. The wellposedness and stability are established for the reduced problem. Our proof is based on the method of energy, the Lax–Milgram lemma, and the inversion theorem of the Laplace transform. Moreover, a priori estimates with explicit dependence on the time are achieved for the quantities of acoustic pressure and elastic displacement by taking special test functions for the timedomain variational problem. (Joint work with Peijun Li and Bo Zhang )
Symplectic RungeKutta methods for Hamiltonian systems driven by rough paths Chuying Huang（黄楚荧）
huangchuying@lsec.cc.ac.cn
Chinese Academy of Sciences
Abstract: We consider Hamiltonian systems driven by multidimensional Gaussian process in rough path sense, with either bounded or linear vector fields. We indicate that the phase flow of this rough Hamiltonian system preserves the symplectic structure, and this property could be inherited almost surely by symplectic RungeKutta methods. Furthermore, we give the solvability condition as well as the convergence rates for RungeKutta methods. Numerical experiments verify our theoretical analysis. This is a joint work with Jialin Hong and Xu Wang.
Dynamical behavior of solutions for the wave equation in inhomogeneous media
Shuguan Ji（冀书关）
Abstract: In this talk, we consider the wave equation with xdependent coefficients, which arises from the vibrations of an inhomogeneous string and the propagation of seismic waves in nonisotropic media. On the one hand, we talk about some results on the existence and multiplicity of the timeperiodic solutions for the conservative nonlinear wave equation with xdependent coefficients. On the other hand, we also consider the dissipative wave equations with xdependent coefficients and give the decay estimates for the energy and the L^2norm of solutions.
Invariant manifold and foliation for random dynamical system and application
Ji Li（李骥）
liji@hust.edu.cn
Huazhong University of Science and Technology
Abstract: 介绍法向双曲不变流形在一致有界扰动下的保持性理论和不变叶层性质。进而将二者应用到含有一致有界随机扰动的奇异摄动系统，建立相关慢流形的保持性质和不变叶层性质。然后在周期震荡及混沌震荡的具体问题中介绍不变流形理论的应用。
Permanence and ergodicity of stochastic GilpinAyala population model
Xiaoyue Li（李晓月）
lixy209@nenu.edu.cn
Northeast Normal University
Abstract: This work is concerned with permanence and ergodicity of stochastic GilpinAyala models involve continuous states as well as discrete events. A distinct feature is that the GilpinAyala parameter and its corresponding perturbation parameter are allowed to be varying randomly in accordance with a random switching process. Necessary and sufficient conditions of the stochastic permanence and extinction are established, which are much weaker than the previous results. The existence of the unique stationary distribution is also established. Our approach treats much wider class of systems, uses much weaker conditions, and substantially generalizes previous results. It is shown that regime switching can suppress the impermanence. Furthermore, several examples and simulations are given to illustrate our main results.
Periodic structure of quasiperiodic system
Zeng Lian（连增）
zenglian@gmail.com
Sichuan University
Abstract: Smale Horseshoe is a classical model which is introduced by Smale in 1960’s to describe the chaotic phenomena of certain dynamical systems. In 1980’s, Katok has shown that for diffeomorphism on compact Riemannian manifolds nonuniformly hyperbolic system persists the existence of infinitely many periodic orbit and Smale horseshoe. However, all of the existing results are for autonomous systems. One natural question is: for nonautonomous systems, is there any special structure which can be viewed as analogue of periodic orbit or horseshoe? In the result I report in this talk, we have defined periodic structure and Smale horseshoe for nonautonomous (or random) systems, and also proved the existence of certain objects for a type of quasiperiodic hyperbolic systems. This work is joint with Wen Huang.
The rate of decay of stable periodic solutions for Duffing equation
with L^pconditions
Shuqing Liang（梁树青）
liangshuqing@jlu.edu.cn
Jilin University
Abstract: For a new class of g(t, x), the existence, uniqueness and stability of 2πperiodic solution of Duffing equation x’’+cx’+g(t,x)=h(t) are presented. Moreover, the unique 2πperiodic solution is (exponentially asymptotically stable) and its rate of exponential decay c/2 is sharp. The new criterion characterizes g’_x(t,x)c^2/4 with Lpnorms (p∈[1,∞]), and the classical criterion employs the L∞norm. The advantage is that we can deal with the case that g’_x(t,x)c^2/4 is beyond the optimal bounds of the L∞norm, because of the difference between the L^pnorm and the L∞norm.
Iterative roots of piecewise monotone functions
Liu Liu（刘鎏）
matliuliu@163.com
Southwest Jiaotong University
Abstract: It has been treated as a difficult problem to find iterative roots of nonmonotonic functions. For some nonmonotonic functions having finitely many forts, a method was given to obtain a nonmonotonic iterative root by extending a monotone iterative root from the characteristic interval. In my talk, I will introduce the results for iterative roots of piecewise monotone functions.
Approximating stochastic partial differential equations with additive
fractional noises
Zhihui Liu（刘智慧）
liuzhihui@lsec.cc.ac.cn
Chinese Academy of Sciences
Abstract: In this talk, we analyze strong convergence rate of Galerkin approximations for stochastic partial differential equations driven by an additive fractional noise which is temporally white and spatially fractional with Hurst index less than or equal to 1/2. First we regularize the noise by the WongZakai approximation and obtain its optimal order of convergence. Then we apply the Galerkin method to discretize the random partial differential equations with regularized noises. Optimal error estimates are obtained for the Galerkin approximations. This is a joint work with Jialin Hong and Liying Zhang.
Stochastic symplectic and multisymplectic methods for stochastic NLS equation with quadratic potential
Lijun Miao（苗利军）
miaolijun@lsec.cc.ac.cn
Chinese Academy of Sciences
Abstract: The Stochastic nonlinear Schrödinger equation with quadratic potential models BoseEinstein condensations under a magnetic trap when $\theta<0$, which is not only an infinitedimensional stochastic Hamiltonian system but also a stochastic Hamiltonian partial differential equation essentially. We firstly indicate that this equation possesses stochastic symplectic structure and stochastic multisymplectic conservation law. It is also shown that the charge and energy have the evolution laws of linear growth with respect to time in the sense of expectation. Moreover, we propose a stochastic symplectic scheme in the temporal discretization, and give the error estimate of this scheme in probability. In order to simulate the evolution laws of charge and energy numerically, we also give a stochastic multisymplectic scheme, of which the corresponding discrete charge and energy are in accordance with the continuous case. Numerical experiments are performed to test the proposed numerical scheme. This is a joint work with Jialin Hong and Liying Zhang.
WongZakai approximations and center manifolds
Jun Shen（申俊）
junshen85@163.com
Sichuan University
Abstract: In this paper, we study the WongZakai approximations given by a stationary process via the Wiener shift and their associated dynamics of the stochastic differential equation driven by a $n$dimensional Brownian motion. We prove that the solutions of WongZakai approximations converge in the mean square to the solutions of the Stratonovich stochastic differential equation. We also show that for a simple multiplicative noise, the centermanifold of the WongZakai approximations converges to the centermanifold of the Stratonovich stochastic differential equation.
High order conformal symplectic and ergodic schemes for stochastic Langevin equation via generating functions
Liying Sun（孙丽莹）
liyingsun@lsec.cc.ac.cn
Chinese Academy of Sciences
Abstract: In this paper, we consider the stochastic Langevin equation with additive noises, which possesses both conformal symplectic geometric structure and ergodicity. We propose a methodology of constructing high weak order conformal symplectic schemes by converting the equation into an equivalent autonomous stochastic Hamiltonian system and modifying the associated generating function. To illustrate this approach, we construct a specific second order numerical scheme, and prove that its symplectic form dissipates exponentially. Moreover, for the linear case, the proposed scheme is also shown to inherit the ergodicity of the original system, and the temporal average of the numerical solution is a proper approximation of the ergodic limit over long time. Numerical experiments are given to verify these theoretical results. This is a joint work with Jialin Hong and Xu Wang.
Minkowski空间中几类具奇性的平均曲率方程径向解的存在性
Libo Wang（王立波）
wlb_math@163.com
Beihua University
Abstract: 利用Leggett–Williams不动点定理，摄动技巧和LeraySchauder度理论等研究Minkowski空间中几类具奇性的平均曲率方程径向解的存在性结果。
Numerical methods for stochastic Hamiltonian systems
Lijin Wang（王丽瑾）
ljwang@lsec.cc.ac.cn
Chinese Academy of Sciences
Abstract: TBA
Asymptotic behavior of nonautonomous random reactiondiffusion equations
on unbounded domains
Xiaohu Wang（王小虎）
wangxiaohu@scu.edu.cn
Sichuan University
Abstract: In this talk, we study the long term behavior of nonautonomous reactiondiffusion equations on unbounded domains with random forcing given by an approximation of white noise. We first prove the existence and uniqueness of tempered pullback attractors for the random reactiondiffusion equations, and then establish the upper semicontinuity of attractors as random forcing approaches white noise, including additive as well as multiplicative noise.
Construction and convergence of an ergodic and conformal multisymplectic approximation for a damped stochastic NLS equation
Xu Wang（王旭）
wangxu@lsec.cc.ac.cn
Chinese Academy of Sciences
Abstract: We propose a novel fully discrete scheme for the damped stochastic nonlinear Schrödinger equation with an additive noise. Theoretical analysis shows that the proposed scheme is of order one in probability under appropriate assumptions for the initial value and noise. Meanwhile, the scheme inherits the ergodicity of the original system with a unique invariant measure, and preserves the discrete conformal multisymplectic conservation law. Numerical experiments are given to show the longtime behavior of the discrete charge and the time average of the numerical solution, and to test the convergence order, which verify our theoretical results. This is a joint work with Jialin Hong and Lihai Ji.
Multiplicity of solutions for nonlocal elliptic equations driven
by the fractional Laplacian
Yuanhong Wei（魏元鸿）
yhwei@amss.ac.cn
Jilin University
Abstract: We consider the semilinear elliptic PDE driven by the fractional Laplacian. By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of nontrivial solutions are established. The validity of the PalaisSmale condition without AmbrosettiRabinowitz condition for nonlocal elliptic equations is proved. Two nontrivial solutions are given under some weak hypotheses. Nonlocal elliptic equations with concaveconvex nonlinearities are also studied, and existence of at least six solutions are obtained. Moreover, a global result of AmbrosettiBrezisCerami type is given, which shows that the effect of the parameter in the nonlinear term changes considerably the nonexistence, existence and multiplicity of solutions.
Near preservation of quadratic invariants via explicit stochastic
Runge–Kutta methods
Liying Zhang（张利英）
lyzhang@lsec.cc.ac.cn
China University of Mining & Technology, Beijing
Abstract: Stochastic Runge–Kutta (SRK) methods preserving quadratic in variants (QIs) of stochastic differential equations (SDEs) are fully implicit generally. Hence iteration methods have to be applied in executing these implicit SRK methods and then the computation cost is expensive in practice. The present work aims to design explicit SRK methods that can nearly pre serve QIs of SDEs to guarantee the good longtime behavior of the numerical solution to some extent. The technique of rooted colored trees is employed to investigate the conditions for the SRK methods to preserve QIs up to any desired order of accuracy almost surely. Combining these conditions with the strong convergence order conditions can offer a practical approach of construct ing explicit SRK methods nearly preserving QIs. Numerical experiments on stochastic Kubo oscillator, stochastic rigid body system I, II and stochastic nonlinear Schrödinger equation are performed. All these experiments show the good longtime stability of the explicit SRK methods that can preserve QIs up to a certain order.
Differentiability of the conjugacy in the HartmanGrobman Theorem
Wenmeng Zhang（张文萌）
mathzwm@sina.com
Chongqing Normal University
Abstract: The classical HartmanGrobman Theorem states that a C1 diffeomorphism near its hyperbolic fixed point is topologically conjugate to its linear part. Usually, improving the smoothness of the conjugacy requires nonresonant conditions and higher smoothness of the system. Otherwise, the conjugacy is not smooth, not even Lipschitz continuous. In this talk, we show that the conjugacy is differentiable at the fixed point without any nonresonant conditions under a very weak smoothness condition.
Optimal control methods for stability criteria on Hill's equations
Jian Zu（祖建）
zuj100@nenu.edu.cn
Northeast Normal University
Abstract: This report gives a tutorial on how to prove Lyapunov type criteria by optimal control methods. Firstly, we consider stability criteria on Hill’s equations with nonnegative potential. By using Wang and Li’s skill developed in 1990s, we obtain several stability criteria including Lyapunov’s criterion and Neigauz and Lidskii’s criterion. Secondly, we present the stability criteria on Hill’s equations with signchanging potential in which Brog’s criterion and Krein’s criterion are included.