Persistence of Lower Dimensional Tori of General Types in Hamiltonian Systems

2000 Mathematics Subject Classification. 37J40.The work is a generalization to [40] in which 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.The first author was partially supported by NSFC grant 19971042, National 973 Project of China: Nonlinearity, the outstanding young's project of Ministry of Education of China, and National outstanding young's award of China. The second author was partially supported by NSF grants DMS9803581 and DMS-0204119

Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows

AMS(MOS) subject classifications: 34C27, 34D05, 35B15, 35B40, 35K57, 54H20.The current series of papers, which consists of three parts, are devoted to the study of almost automorphic dynamics in differential equations. By making use of techniques from abstract topological dynamics, we show that almost automorphy, a notion which was introduced by S. Bochner in 1955, is essential and fundamental in the qualitative study of almost periodic differential equations. Fundamental notions from topological dynamics are introduced in the first part. Harmonic properties of almost automorphic functions such as Fourier series and frequency module are studied. A module containment result is provided. In the second part, we study lifting dynamics of w-limit sets and minimal sets of a skew-product semiflow from an almost periodic minimal base flow. Skewproduct semiflows with (strongly) order preserving or monotone natures on fibers are given a particular attention. It is proved that a linearly stable minimal set must be almost automorphic and become almost periodic if it is also uniformly stable. Other issues such as flow extensions and the existence of almost periodic global attractors, etc. are also studied. The third part of the series deals with dynamics of almost periodic differential equations. In this part, we apply the general theory developed in the previous two parts to study almost automorphic and almost periodic dynamics which are lifted from certain coefficient structures (e.g., almost automorphic or almost periodic) of differential equations. It is shown that (harmonic or subharmonic) almost automorphic solutions exist for a large class of almost periodic ordinary, parabolic and delay differential equations.Partially supported by NSF grants DMS-9207069, DMS-9402945 and DMS-9501412

Quasi-Periodic Solutions in a Nonlinear Schrödinger Equation

1991 Mathematics Subject Classification. Primary 37K55, 35B10, 35J10, 35Q40, 35Q55.In this paper, one-dimensional (1D) nonlinear Schrödinger equation [equation omitted] with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer N > 1, the equation admits a Whitney smooth family of small-amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkho ff normal form reduction and an improved KAM method.Partially supported by NSF grant DMS0204119

Asymptotic pairs, stable sets and chaos in positive entropy systems

We consider positive entropy $G$-systems for certain countable, discrete, infinite left-orderable amenable groups $G$. By undertaking local analysis, the existence of asymptotic pairs and chaotic sets will be studied in connecting with the stable sets. Examples are given for the case of integer lattice groups, the Heisenberg group, and the groups of integral unipotent upper triangular matrices

Quasi-Periodic Breathers in Hamiltonian Networks of Long-Range Coupling

1991 Mathematics Subject Classification. Primary 37K60, 37K55This work is concerned with Hamiltonian networks of weakly and long-range coupled oscillators with either variable or constant on-site frequencies. We derive an infinite dimensional KAM-like theorem by which we establish that, given any N-sites of the lattice, there is a positive measure set of small amplitude, quasi-periodic breathers (solutions of the Hamiltonian network that are quasi-periodic in time and exponentially localized in space) having N-frequencies which are only slightly deformed from the on-site frequencies.The first author is partially supported by NSFC grant 10771098, 973 projects of China and NSFJP grant BK2007134. The third author is partially supported by NSFC grant 10428101 and NSF grants DMS0204119, DMS0708331

Metapopulation Dynamics with Migration and Local Competition

MSC2000: primary 34D15, 34D23, 92B05; secondary 34K60, 34E13, 92D40.Many patch-based metapopulation models assume that the local population within each patch is at its equilibrium and independent of changes in patch occupancy. We study a metapopulation model which explicitly incorporates the local population dynamics of two competing species. Singular perturbation method is used to separate the fast dynamics of the local competition and the slow process of patch colonization and extinction. Our results show that the coupled system leads to much more complex outcomes than simple patch models that do not include explicit local dynamics.The first author was partially supported by NSF grant DMS-9974389 and NSF grant ESE-0119908. The second author was partially supported by NSF grant DMS-9803581. The third author was partially supported by a Canadian NSERC postdoctoral fellowship

On Almost Automorphic Dynamics in Symbolic Lattices

1991 Mathematics Subject Classification. Primary Primary 37B10, 37A35, 43A60; Secondary 37B20, 54H20.We study the existence, structure, and topological entropy of almost automorphic arrays in symbolic lattice dynamical systems. In particular we show that almost automorphic arrays with arbitrarily large entropy are typical in symbolic lattice dynamical systems. Applications to pattern formation and spatial chaos in infinite dimensional lattice systems are considered, and the construction of chaotic almost automorphic signals is discussed.The first author was supported by a Max Kade Postdoctoral Fellowship (at Georgia Tech). The second author was partially supported by DFG grant Si 801 and CDSNS, Georgia Tech. The third author was partially supported by NSF Grant DMS-0204119