Xiao-Biao Lin

Upper semicontinuity of attractors for approximations of semigroups and partial differential equations

Abstract : Suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system. Conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined. Applications are given to parabolic and hyperbolic partial differential equations.

Smooth Invariant Foliations in Infinite Dimensional Spaces

One of the most useful properties of dynamical systems is the existence of invariant manifolds and their invariant foliations near an equilibrium or a periodic orbit. These manifolds and foliations serve as a convenient setting to describe the qualitative behavior of the local flows, and in many cases they are useful tools for technical estimates which facilitate the study of the local bifurcation diagram (see [6]). Many other important concepts in dynamical systems are closely related to the invariant manifolds and foliations. In finite dimensional space, the relations among invariant manifolds, invariant foliations, l-lemma, linearization, and homoclinic bifurcation have been studied in [ll]. It is well known that if each leaf is used as a coordinate, the original system is completely decoupled and the linearization follows easily (for example, see [27, 221). As a motivation, let us consider a linear system in Rmfn

Heteroclinic orbits for retarded functional differential equations

Abstract Suppose Γ is a heteroclinic orbit of a Ck functional differential equation x (t) = ƒ(x i ) with α-limit set α(Γ) and ω-limit set ω(Γ) being either hyperbolic equilibrium points or periodic orbits. Necessary and sufficient conditions are given for the existence of an ftf close to ƒ in Ck with the property that x (t) = \ tf(x t ) has a heteroclinic orbit \ gG close to Γ. The orbits \ gG are obtained from the zeros of a finite number of bifurcation functions G(β, f )∈R d ∗ ,β ∈R d + 1 . Transversality of Γ is characterized by the nondegeneracy of the derivative DβG. It is also shown that the \ tf which have heteroclinic orbits near Γ are on a Ck submanifold of finite codimension = max{0, − indΓ} or on the closure of it, where ind Γ is the index of Γ.

Exponential dichotomies and homoclinic orbits in functional differential equations

Abstract Suppose an autonomous functional differential equation has an orbit Γ which is homoclinic to a hyperbolic equilibrium point. The purpose of this paper is to give a procedure for determining the behavior of the solutions near Γ of a functional differential equation which is a nonautonomous periodic perturbation of the original one. The procedure uses exponential dichotomies and the Fredholm alternative. It is also shown that any smooth function p ( t ) defined on the reals which approaches zero monotonically as t → ± ∞ is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero. Examples illustrating the results are also given.

Symbolic dynamics and nonlinear semiflows

SummaryFor a transverse homoclinic orbit γ of a mapping (not necessarily invertible) on a Banach space, it is shown that the mapping restricted to orbits near γ is equivalent to the shift automorphism on doubly infinite sequences on finitely many symbols. Implications of this result for the Poincaré map of semiflows are given.

Traveling Wave Solutions for a Predator–Prey System With Sigmoidal Response Function

We study the existence of traveling wave solutions for a diffusive predator–prey system. The system considered in this paper is governed by a Sigmoidal response function which in some applications is more realistic than the Holling type I, II responses, and more general than a simplified form of the Holling type III response considered before. Our method is an improvement to the original method introduced in the work of Dunbar (J Math Biol 17:11–32, 1983; SIAM J Appl Math 46:1057–1078, 1986). A bounded Wazewski set is used in this work while unbounded Wazewski sets were used in Dunbar (1983, 1986). The existence of traveling wave solutions connecting two equilibria is established by using the original Wazewski’s theorem which is much simpler than the extended version in Dunbar’s work.

A shadowing lemma with applications to semilinear parabolic equations

The property of hyperbolic sets that is embodied in the Shadowing Lemma is of great importance in the theory of dynamical systems. In this paper a new proof of the lemma is presented, which applies not only to the usual case of a diffeomorphism in finite-dimensional space but also to a sequence of possibly noninvertible maps in a Banach space. The approach is via Newton’s method, the main step being the verification that a certain linear operator is invertible. At the end of the paper an application to parabolic evolution equations is given.

Shadowing lemma and singularly perturbed boundary value problems

A complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in

Heteroclinic Bifurcation and Singularly Perturbed Boundary Value Problems

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Transition Layers for Singularly Perturbed Delay Differential Equations with Monotone Nonlinearities

. The validity of such expansions is deduced from a gene...