Kening Lu

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

We consider a one-dimensional lattice with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and additive independent white noise at each node. We prove the existence of a compact global random attractor within the set of tempered random bounded sets. An interesting feature of this is that, even though the spatial domain is unbounded and the solution operator is not smoothing or compact, pulled back bounded sets of initial data converge under the forward flow to a random compact invariant set.

ATTRACTORS FOR LATTICE DYNAMICAL SYSTEMS

We study the asymptotic behavior of solutions for lattice dynamical systems. We first prove asymptotic compactness and then establish the existence of global attractors. The upper semicontinuity of the global attractor is also obtained when the lattice differential equations are approached by finite-dimensional systems.

Existence and persistence of invariant manifolds for semiflows in Banach space

Introduction Notation and preliminaries Statements of theorems Local coordinate systems Cone lemmas Center-unstable manifold Center-stable manifold Smoothness of center-stable manifold Smoothness of center-unstable manifold Persistence of invariant manifold Persistence of normal hyperbolicity Invariant manifolds for perturbed semiflow References.

Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perron’s method. Then, we prove the smoothness of these invariant manifolds.

Invariant foliations near normally hyperbolic invariant manifolds for semiflows

Let M be a compact C1 manifold which is invariant and normally hyperbolic with respect to a C1 semiflow in a Banach space. Then in an -neighborhood of M there exist local C1 center-stable and center-unstable manifolds W cs( ) and W cu( ), respectively. Here we show that W cs( ) and W cu( ) may each be decomposed into the disjoint union of C1 submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects M in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.

Persistence of Overflowing Manifolds for Semiflow

This paper, which is a sequel to a previous one [4] by the same authors, is devoted to the persistence of overflowing manifolds and inflowing manifolds for a semiflow in a Banach space. We consider a C1 semiflow defined on a Banach space X; that is, it is continuous on [0,∞)×X , and for each t ≥ 0, T t : X → X is C1, and T t ◦ T (x) = T t+s(x) for all t, s ≥ 0 and x ∈ X . A typical example is the solution operator for a differential equation. In [4] we proved that a compact, normally hyperbolic, invariant manifold M persists under small C1 perturbations in the semiflow. We also showed that in a neighborhood of M , there exist a center-stable manifold and a center-unstable manifold that intersect in the manifold M . In [4] the compactness and invariance of the manifold M were important assumptions. In the present paper, we study the more general case where the manifold M is overflowing (“negatively invariant and the semiflow crosses the boundary transversally”) or inflowing (“positively invariant and the semiflow crosses the boundary transversally”). We do not assume that M is compact or finite-dimensional. Also, M is not necessarily an imbedded manifold, but an immersed manifold. As an example, a local unstable manifold of an equilibrium point is an overflowing manifold. In brief, our main results on the overflowing manifolds may be summarized as follows (the precise statements are given in Section 2). We assume that the immersed manifoldM does not twist very much locally,M is covered by the image under T t of a subset a positive distance away from boundary, DT t contracts along the normal direction and does so more strongly than it does along the tangential direction, and DT t has a certain uniform continuity in a neighborhood of M . If the C1 perturbation T t of T t is sufficiently close to T t, then T t has a unique C1 immersed overflowing manifold M nearM . Furthermore, if T t isCk and a spectral gap condition holds, then M isCk. Similar results for inflowing manifolds are also obtained and given in Section 7.

Invariant manifolds for random and stochastic partial differential equations

Abstract Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic infuences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo- stable and pseudo-unstable manifolds for a class of random partial differential equations and stochastic partial differential equations is shown. Unlike the in- variant manifold theory for stochastic ordinary differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with linear multiplicative noise.

A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations

In this paper, we study the structural stability of the Cahn-Hilliard equation and the phase-field equations. We show that the Cahn-Hilliard equation and the phase-field equations are topologically conjugate to a decoupled system of a linear equation of infinite dimension and an ordinary differential equation which is the reduced equation on the inertial manifold; particularly, the flow nearby hyperbolic stationary solutions is structurally stable.

The period function of hyperelliptic Hamiltonians of degree 5 with real critical points

We provide a criterion to determine the convexity of the period function for a class of planar Hamiltonian systems. As an application we prove that the period function of all hyperelliptic Hamiltonians of degree 5 with real critical points has at most one simple critical point. More precisely, if the period annulus surrounds only one non-degenerate singularity, then the period function is monotone; otherwise (i.e. the period annulus surrounds three singularities, taking into account the multiplicities), the period function has exactly one critical point.

Rotation numbers for random dynamical systems on the circle

In this paper, we study rotation numbers of random dynamical systems on the circle. We prove the existence of rotation numbers and the continuous dependence of rotation numbers on the systems. As an application, we prove a theorem on analytic conjugacy to a circle rotation.