Person:

Chow, Shui-Nee

Permanent Link

https://hdl.handle.net/1853/71155

Associated Organization(s)

Organizational Unit

School of Mathematics

Full item page

Publication Search Results

Now showing 1 - 9 of 9

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

(Georgia Institute of Technology, 2005-07-23) van Noort, Martijn ; Porter, Mason ; Yi, Yingfei ; Chow, Shui-Nee

We employ KAM theory to rigorously investigate the transition between quasiperiodic and chaotic 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 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 its amplitude essentially only affects scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.
Quasiperiodic Dynamics in Hamiltonian 1 1/2 Degree of Freedom Systems Far from Integrability

(Georgia Institute of Technology, 2005-01-21) Chow, Shui-Nee ; van Noort, Martijn ; Yi, Yingfei

The subject of this paper is two-quasiperiodicity in a large class of one-and-a-half degree of freedom Hamiltonian systems. The main result is that such systems have invariant tori for any internal frequency that is of constant type and sufficiently large, relative to the forcing frequency. An explicit bound on the minimum value of the internal frequency is presented. The systems under consideration are not required to be small perturbations of integrable ones.
Travelling Wave Solutions in a Tissue Interaction Model for Skin Pattern Formation

(Georgia Institute of Technology, 2003) Ai, Shangbing ; Chow, Shui-Nee ; Yi, Yingfei

We discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction model on skin pattern formation proposed by Cruywagen and Murray. The geometric theory of singular perturbations is employed.
The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian System with Elliptic Segment

(Georgia Institute of Technology, 2000) Chow, Shui-Nee ; Li, Chengzhi ; Yi, Yingfei

We study the cyclicity of period annuli (or annulus) for general degenerate quadratic Hamiltonian systems with an elliptic segment or a saddle loop, under quadratic perturbations. By using geometrical arguments and studying the respective Abelian integral based on the Picard-Fuchs equation, it is shown that the cyclicity of period annuli or annulus for such systems equals two. This result, together with those of [8],[10],[11],[18],[19], gives a complete solution to the infinitesimal Hilbert 16th problem in the case of degenerate quadratic Hamiltonian systems under quadratic perturbations.
Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

(Georgia Institute of Technology, 1999) Chow, Shui-Nee ; Li, Yong ; Yi, Yingfei

Generalizing the degenerate KAM theorem under the Rüssmann non-degeneracy and the isoenergetic KAM theorem, we employ a quasi-linear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth sub-manifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the sub-manifold, we shall show the following: a) the majority of the unperturbed tori on the sub-manifold 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 sub-isoenergetic 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 Manifolds for Invariant Sets

(Georgia Institute of Technology, 1999) Chow, Shui-Nee ; Liu, Weishi ; Yi, Yingfei

We derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector fields in R^n. By applying the Hadamard graph transform technique, it is shown that, associated to certain dynamical characteristics of the linearized flow along the invariant set, there exists an invariant manifold (called a center manifold) of the invariant set which contains every locally bounded solution (in particular, contains the invariant set) and is persistent under small perturbations.
Center Manifolds for Invariant Manifolds

(Georgia Institute of Technology, 1997) Chow, Shui-Nee ; Liu, Weishi ; Yi, Yingfei

We study dynamics of flows generated from smooth vector fields in R^n in the vicinity of an invariant and closed smooth manifold Y. By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of Y) based on the information of the linearization along Y, which contains every locally bounded solution and is persistent under small perturbations.
Stability and Bifurcation of Traveling Wave Solutions in Coupled Map Lattices

(Georgia Institute of Technology, 1994) Chow, Shui-Nee ; Shen, Wenxian

A class of coupled map lattices are considered. For each such coupled map lattice, the stability and bifurcation of its traveling wave solutions are studied. A stability criteria is given. It is shown that bifurcation scenario in coupled map lattices is similar to but different from that arising in reaction diffusion equations. As an application of the general results, the dynamics near traveling wave solutions in a discrete Huxley equation is discussed.
Dynamics in a Discrete Nagumo Equation - Spatial Topological Chaos

(Georgia Institute of Technology, 1994) Chow, Shui-Nee ; Shen, Wenxian

In this paper we show the existence of spatial topological chaos in a discrete Nagumo equation, and discuss the similarities and differences between the discrete Nagumao equation and the continuous one.