A note on partially hyperbolic attractors: Entropy conjecture and SRB measures
Peidong Liu Kening Lu
Discrete & Continuous Dynamical Systems - A 2015, 35(1): 341-352 doi: 10.3934/dcds.2015.35.341
In this note we show that, for a class of partially hyperbolic $C^r$ ($r \geq 1$) diffeomorphisms, (1) Shub's entropy conjecture holds true; (2) SRB measures exist as zero-noise limits.
keywords: Partially hyperbolic attractors random perturbations. entropy conjecture SRB measures
Jibin Li Kening Lu Junping Shi Chongchun Zeng
Discrete & Continuous Dynamical Systems - A 2009, 24(3): i-ii doi: 10.3934/dcds.2009.24.3i
This special issue of Discrete and Continuous Dynamical Systems-A is dedicated to Peter W. Bates on the occasion of his 60th birthday, and in recognition of his outstanding contributions to infinite dimensional dynamical systems and the mathematical theory of phase transitions.
    Peter Bates was born in Manchester, England on December 27, 1947. He graduated from the University of London in mathematics in 1969 after which he moved to United States with his family. Later, he attended the University of Utah and received his Ph.D. in 1976. Following his graduation, Peter moved to Texas and taught at University of Texas at Pan American and Texas A&M University. He returned to Utah in 1984 and taught at Brigham Young University until 2004. He is currently a professor of mathematics at Michigan State University.

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Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains
Dingshi Li Kening Lu Bixiang Wang Xiaohu Wang
Discrete & Continuous Dynamical Systems - A 2018, 38(1): 187-208 doi: 10.3934/dcds.2018009

In this paper, we study the limiting behavior of dynamics for stochastic reaction-diffusion equations driven by an additive noise and a deterministic non-autonomous forcing on an (n+1)-dimensional thin region when it collapses into an n-dimensional region. We first established the existence of attractors and their properties for these equations on (n+1)-dimensional thin domains. We then show that these attractors converge to the random attractor of the limit equation under the usual semi-distance as the thinness goes to zero.

keywords: Thin domain stochastic reaction-diffusion equation pullback attractor upper semicontinuity
A Siegel theorem for dynamical systems under random perturbations
Weigu Li Kening Lu
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 635-642 doi: 10.3934/dcdsb.2008.9.635
In this paper, we generalize the classical Siegel’s theorem for deterministic dynamical systems to that under random perturbations.
keywords: Random dynamical systems Siegel's theorem.
Takens theorem for random dynamical systems
Weigu Li Kening Lu
Discrete & Continuous Dynamical Systems - B 2016, 21(9): 3191-3207 doi: 10.3934/dcdsb.2016093
In this paper, we study random dynamical systems with partial hyperbolic fixed points and prove the smooth conjugacy theorems of Takens type based on their Lyapunov exponents.
keywords: normal forms. random dynamical systems Multiplicative Ergodic theorem Takens' theorem
Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion
María J. Garrido–Atienza Kening Lu Björn Schmalfuss
Discrete & Continuous Dynamical Systems - B 2010, 14(2): 473-493 doi: 10.3934/dcdsb.2010.14.473
In this paper we study nonlinear stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion (fBm) with the Hurst parameter bigger than $1/2$. We show that these SPDEs generate random dynamical systems (or stochastic flows) by using the stochastic calculus for an fBm where the stochastic integrals are defined by integrands given by fractional derivatives. In particular, we emphasize that the coefficients in front of the fractional noise are non-trivial.
keywords: random dynamical systems. Stochastic PDEs fractional Brownian motion
Random attractors for stochastic parabolic equations with additive noise in weighted spaces
Xiaojun Li Xiliang Li Kening Lu
Communications on Pure & Applied Analysis 2018, 17(3): 729-749 doi: 10.3934/cpaa.2018038

In this paper, we establish the existence of random attractors for stochastic parabolic equations driven by additive noise as well as deterministic non-autonomous forcing terms in weighted Lebesgue spaces $ L_{\delta}^r(\mathcal{O})$, where $ 1<r<\infty ,\ \delta$ is the distance from $ x$ to the boundary. The nonlinearity $ f(x,u)$ of equation depending on the spatial variable does not have the bound on the derivative in $ u$, and then causes critical exponent. In both subcritical and critical cases, we get the well-posedness and dissipativeness of the problem under consideration and, by smoothing property of heat semigroup in weighted space, the asymptotical compactness of random dynamical system corresponding to the original system.

keywords: Stochastic parabolic equation random attractor weighted space heat semigroup
Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$
María J. Garrido–Atienza Kening Lu Björn Schmalfuss
Discrete & Continuous Dynamical Systems - B 2015, 20(8): 2553-2581 doi: 10.3934/dcdsb.2015.20.2553
In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting of a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which in particular includes white noise.
keywords: Stochastic PDEs Hilbert-space valued fractional Brownian motion pathwise solutions.
Normal forms for quasiperiodic evolutionary equations
Shui-Nee Chow Kening Lu Yun-Qiu Shen
Discrete & Continuous Dynamical Systems - A 1996, 2(1): 65-94 doi: 10.3934/dcds.1996.2.65
In this paper, we study the normal forms and analytic conjugacy for a class of analytic quasiperiodic evolutionary equations including parabolic equations and Schrödinger equations. We first obtain a normal form theory. Then as a special case of the normal form theory, we show that if the frequency and the eigenvalues satisfy certain small divisor conditions then the nonlinear equation is locally analytically conjugated to a linear equation. In other words, the normal form is a linear equation.
keywords: quasiperiodic evolutionary equations parabolic equations and Schrödinger equations.
Invariant foliations for random dynamical systems
Ji Li Kening Lu Peter W. Bates
Discrete & Continuous Dynamical Systems - A 2014, 34(9): 3639-3666 doi: 10.3934/dcds.2014.34.3639
We prove the existence of invariant foliations of stable and unstable manifolds of a normally hyperbolic random invariant manifold. The normally hyperbolic random invariant manifold referred to here is that which was shown to exist in a previous paper when a deterministic dynamical system having a normally hyperbolic invariant manifold is subjected to a small random perturbation.
keywords: random normally hyperbolic invariant manifolds Random dynamical systems random invariant foliations.

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