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# Workshop on dynamical system and application

2017-05-10

Workshop on dynamical system and application

Organizer: School of Mathematical Sciences, Dalian University of Technology

Time: 11-14 May, 2017

Location: Dalian Xinghai Golf Hotel

Contact: Prof. LIU Zhenxin

Workshop Program

 May 12 Time Speaker Talk Title 8:10-8:30 Opening (Yufeng Lu), Taking photos Chair: Yong Li 8:30-9:00 Zeng  Lian Periodic  structure of quasi-periodic system 9:00-9:30 Wenmeng  Zhang Differentiability  of the conjugacy in the Hartman-Grobman Theorem 9:30-10:00 Jun  Shen Wong-Zakai approximations and center  manifolds 10:00-10:20 Tea break Chair: Fuzhong Cong 10:20-10:50 Lijin  Wang Numerical  methods for stochastic Hamiltonian systems 10:50-11:20 Yuanhong  Wei Multiplicity  of solutions for  non-local elliptic equations driven by the fractional  Laplacian 11:20-11:50 Xu  Wang Construction  and convergence of an ergodic and conformal multi-symplectic approximation  for a damped stochastic NLS equation 12:00 Lunch Chair: Jialin Hong 14:00-14:30 David  Cheban Levitan almost periodic solutions for  stochastic differential equations 14:30-15:00 Xiaohu  Wang Asymptotic  behavior of non-autonomous random reaction-diffusion equations on unbounded  domains 15:00-15:30 Jian  Zu Optimal  control methods for stability criteria on Hill's equations 15:30-15:50 Tea break Chair: Shaoyun Shi 15:50-16:20 Xingwu  Chen Restricted  independence in displacement function for estimating cyclicity 16:20-16:50 Shuqing  Liang The  rate of decay of stable periodic solutions for Duffing equation with L^p-conditions 16:50-17:20 Liying  Zhang Near  preservation of quadratic invariants via explicit stochastic Runge–Kutta  methods 17:20-17:50 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 8:00-8:30 Shuguan  Ji Dynamical  behavior of solutions for the wave equation in inhomogeneous media 8:30-9:00 Yixian  Gao Analysis  of transient acoustic-elastic interaction in an unbounded structure 9:00-9:30 Peng Gao Strong  averaging principle for stochastic Klein-Gordon equation with a fast  oscillation 9:30-10:00 Lijun  Miao Stochastic symplectic and  multi--symplectic methods for stochastic NLS equation with quadratic  potential 10:00-10:20 Tea break Chair: Shaoli Wang 10:20-10:50 Xiaoyue  Li Permanence and  ergodicity of stochastic Gilpin-Ayala population model 10:50-11:20 Rui Gao Decay  of correlations for Fibonacci unimodal interval maps 11:20-11:50 Zhihui Liu Approximating  stochastic partial differential equations with additive fractional noises 12:00 Lunch Chair: Wen Huang 14:00-14:30 Ji  Li Invariant  manifold and foliation for random dynamical system and application 14:30-15:00 Feng  Chen Almost  automorphic solutions of mean field stochastic differential equations 15:00-15:30 Liying  Sun High order conformal symplectic and  ergodic schemes for stochastic Langevin equation via generating functions 15:30-15:50 Tea break Chair: Fengqi Yi 15:50-16:20 Guanggan  Chen Approximating  dynamics of a singularly perturbed stochastic wave equation with a random  dynamical boundary condition 16:50-16:50 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 16:50-17:20 Liu  Liu Iterative  roots of piecewise monotone functions 17:20-17:50 Chuying  Huang Symplectic  Runge-Kutta 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 semi-linear 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

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

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

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 center-focus 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 center-focus 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

jianbocui@lsec.cc.ac.cn

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 Crank--Nicolson 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 Klein-Gordon equation

with a fast oscillation

Peng Gao

gaopengjilindaxue@126.com

Northeast Normal University

Abstract: This paper investigates an averaging principle for stochastic Klein-Gordon equation with a fast oscillation arising as the solution of a stochastic reaction-diffusion 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 time-scales. To be more precise, the well-posedness of mild solutions of the stochastic hyperbolic-parabolic equations is firstly established by applying the fixed point theorem and the cut-off 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 Klein-Gordon 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

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 $s_n=s_{n-1}+\kappa s_{n-2}$ 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 acoustic-elastic interaction in an unbounded structure

Yixian Gao

gaoyx643@nenu.edu.cn

Northeast Normal University

Abstract: Consider the wave propagation in a two-layered 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 time-domain analysis of such an acoustic-elastic 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 well-posedness 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 time-domain variational problem.  (Joint work with Peijun Li and Bo Zhang )

Symplectic Runge-Kutta methods for Hamiltonian systems driven by rough paths Chuying Huang

huangchuying@lsec.cc.ac.cn

Abstract: We consider Hamiltonian systems driven by multi-dimensional 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 Runge-Kutta methods. Furthermore, we give the solvability condition as well as the convergence rates for Runge-Kutta 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

Abstract: In this talk, we consider the wave equation with x-dependent 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 time-periodic solutions for the conservative nonlinear wave equation with x-dependent coefficients. On the other hand, we also consider the dissipative wave equations with x-dependent coefficients and give the decay estimates for the energy and the L^2-norm of solutions.

Invariant manifold and foliation for random dynamical system and application

Ji Li

liji@hust.edu.cn

Huazhong University of Science and Technology

Abstract: Introduce the conservation theory and invariant leaf property of normal hyperbolic invariant manifolds under uniform bounded perturbation. Then apply them to singular perturbation systems with uniformly bounded random perturbations and establish the conservation theory and invariant leaf property for slow manifold. Finally, introduce the applications of invariant manifold theory in the specific problems of periodic shock.

Permanence and ergodicity of stochastic Gilpin-Ayala population model

Xiaoyue Li

lixy209@nenu.edu.cn

Northeast Normal University

Abstract: This work is concerned with permanence and ergodicity of stochastic Gilpin-Ayala models involve continuous states as well as discrete events. A distinct feature is that the Gilpin-Ayala 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 quasi-periodic system

Zeng Lian

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 non-autonomous 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 non-autonomous (or random) systems, and also proved the existence of certain objects for a type of quasi-periodic hyperbolic systems. This work is joint with Wen Huang.

The rate of decay of stable periodic solutions for Duffing equation

with L^p-conditions

Shuqing Liang

liangshuqing@jlu.edu.cn

Jilin University

Abstract: For a new class of g(t, x), the existence, uniqueness and stability of 2π-periodic solution of Duffing equation x’’+cx’+g(t,x)=h(t) are presented. Moreover, the unique 2π-periodic 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 Lp-norms (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^p-norm and the L∞-norm.

Iterative roots of piecewise monotone functions

Liu Liu

matliuliu@163.com

Southwest Jiaotong University

Abstract: It has been treated as a difficult problem to find iterative roots of non-monotonic functions. For some non-monotonic functions having finitely many forts, a method was given to obtain a non-monotonic 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

liuzhihui@lsec.cc.ac.cn

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 Wong-Zakai 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 multi-symplectic methods for stochastic NLS equation with quadratic potential

Lijun Miao

miaolijun@lsec.cc.ac.cn

Abstract: The Stochastic nonlinear Schrödinger equation with quadratic potential models Bose-Einstein condensations under a magnetic trap when $\theta<0$, which is not only an infinite-dimensional stochastic Hamiltonian system but also a stochastic Hamiltonian partial differential equation essentially. We firstly indicate that this equation possesses stochastic symplectic structure and stochastic multi-symplectic 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 multi-symplectic 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.

Wong-Zakai approximations and center manifolds

Jun Shen

junshen85@163.com

Sichuan University

Abstract: In this paper, we study the Wong-Zakai 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 Wong-Zakai 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 center-manifold of the Wong-Zakai approximations converges to the center-manifold of the Stratonovich stochastic differential equation.

High order conformal symplectic and ergodic schemes for stochastic Langevin equation via generating functions

Liying Sun

liyingsun@lsec.cc.ac.cn

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.

Existence of Radial Solutions for Some Singular Mean Curvature Equations in Minkowski Space

Libo Wang

wlb_math@163.com

Beihua University

Abstract: Use Leggett–Williams fixed point theorem, perturbation method and Leray-Schauder degree and other theories to research the existence of Radial Solutions for Some Singular Mean Curvature Equations in Minkowski Space.

Numerical methods for stochastic Hamiltonian systems

Lijin Wang

ljwang@lsec.cc.ac.cn

Abstract: TBA

Asymptotic behavior of non-autonomous random reaction-diffusion equations

on unbounded domains

Xiaohu Wang

wangxiaohu@scu.edu.cn

Sichuan University

Abstract: In this talk, we study the long term behavior of non-autonomous reaction-diffusion 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 reaction-diffusion 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 multi-symplectic approximation for a damped stochastic NLS equation

Xu Wang

wangxu@lsec.cc.ac.cn

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 multi-symplectic 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 non-local elliptic equations driven

by the fractional Laplacian

Yuanhong Wei

yhwei@amss.ac.cn

Jilin University

Abstract: We consider the semi-linear elliptic PDE driven by the fractional Laplacian. By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of non-trivial solutions are established. The validity of the Palais-Smale condition without Ambrosetti-Rabinowitz condition for non-local elliptic equations is proved. Two non-trivial solutions are given under some weak hypotheses. Non-local elliptic equations with concave-convex nonlinearities are also studied, and existence of at least six solutions are obtained. Moreover, a global result of Ambrosetti-Brezis-Cerami 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

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 long-time 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 long-time stability of the explicit SRK methods that can preserve QIs up to a certain order.

Differentiability of the conjugacy in the Hartman-Grobman Theorem

Wenmeng Zhang

mathzwm@sina.com

Chongqing Normal University

Abstract: The classical Hartman-Grobman 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 non-resonant 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 non-resonant conditions under a very weak smoothness condition.

Optimal control methods for stability criteria on Hill's equations

Jian Zu

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 sign-changing potential in which Brog’s criterion and Krein’s criterion are included.