ngfei Yi

Almost automorphic and almost periodic dynamics in skew-product semiflows

Acknowledgment Abstract Almost automorphy and almost periodicity Skew-product semiflows Applications to differential equations.

CENTER MANIFOLDS FOR SMOOTH INVARIANT MANIFOLDS

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9947-00-02443-0. First published in Trans. Amer. Math. Soc. in 2000, published by the American Mathematical Society.

Relaxation Oscillations in a Class of Predator-Prey Systems

Abstract We consider a class of three-dimensional, singularly perturbed predator–prey systems having two predators competing exploitatively for the same prey in a constant environment. By using dynamical systems techniques and the geometric singular perturbation theory, we give precise conditions which guarantee the existence of stable relaxation oscillations for systems within the class. Such result shows the coexistence of the predators and the prey with quite diversified time response which typically happens when the prey population grows much faster than those of predators. As an application, a well-known model will be discussed in detail by showing the existence of stable relaxation oscillations for a wide range of parameters values of the model.

On minimal sets of scalar parabolic equations with skew-product structures

Skew-product semi-flow I-t: X x Y -+ X x Y which is generated by Ut -= Uxx + f(Y*t Ix, u, ux), t>0, 0<x< 1, y e Y, D or N boundary conditions is considered, where X is an appropriate subspace of H2(0, 1), (Y, R) is a minimal flow with compact phase space. It is shown that a minimal set E C X x Y of Ilt is an almost 1-1 extension of Y, that is, set Yo = {y E Y I card(E c P 1 (y)) = I} is a residual subset of Y, where P: X x YY is the natural projection. Consequently, if (Y, R) is almost periodic minimal, then any minimal set E C X x Y of Ilt is an almost automorphic minimal set. It is also proved that dynamics of [It is closed in the category of almost automorphy, that is, a minimal set E C X x Y of FIt is almost automorphic minimal if and only if (Y, R) is almost automorphic minimal. Asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed. Examples of nonalmost periodic almost automorphic minimal sets are provided.

Positive solutions of super-critical elliptic equations and asymptotics

This paper is devoted to the study of positive solutions of semilinear elliptic equations . Asymp- totic behavior of ground states and uniqueness of singular ground states are proved via invariant manifold theory of dynamical systems. The Dirichlet problem in exterior domains is also studied. It is proved that this problem has one positive solution with fast decay and infinitely many positive solutions with slow decay. The asymptotics of the singular sequence of fast decay solutions when p approaches to is also discussed.

Persistence of Lower Dimensional Tori of General Types in Hamiltonian Systems

This work is a generalization to a result of J. You (1999) We study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.

Persistence of Invariant Tori in Generalized Hamiltonian Systems

We present some results of KAM type, comparable to the KAM theory for volume-preserving maps and flows, for generalized Hamiltonian systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd dimensional. Applications to the perturbation of three-dimensional steady Euler fluid particle path flows are considered with respect to the existence problem of barriers to fluid transport and mixing.

Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

SummaryGeneralizing the degenerate KAM theorem under the Rüssmann nondegeneracy and the isoenergetic KAM theorem, we employ a quasilinear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth submanifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the submanifold, we shall show the following: (a) the majority of the unperturbed tori on the submanifold will persist; (b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; (c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a subisoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.

Center manifold and stability for skew-product flows

We study the existence and smoothness of global center, center-stable, and center-unstable manifolds for skew-product flows. Smooth invariant foliations to the center stable and center unstable manifolds are also discussed.

Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices

We employ KAM theory to rigorously investigate quasiperiodic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices. Toward this end, we apply a coherent structure ansatz to the Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation describing the spatial dynamics of the condensate. For shallow-well, intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather sets to prove that one obtains mostly quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, where the minimal amplitude depends on the experimentally adjustable BEC parameters. We show that this threshold scales with the square root of the inverse of the two-body scattering length, whereas the rotation number of tori above this threshold is proportional to the amplitude. As a consequence, one obtains the same dynamical picture for lattices of all depths, as an increase in depth essentially affects only scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.