Chaotic Field Theory - a Sketch

Spatio-temporally chaotic dynamics of a classical field can be described by means of an infinite hierarchy of its unstable spatio-temporally periodic solutions. The periodic orbit theory yields the global averages characterizing the chaotic dynamics, as well as the starting semiclassical approximation to the quantum theory. New methods for computing corrections to the semiclassical approximation are developed; in particular, a nonlinear field transformation yields the perturbative corrections in a form more compact than the Feynman diagram expansions.Comment: 22 pp, 24 figs, uses elsart.cl

Turbulent fields and their recurrences

We introduce a new variational method for finding periodic orbits of flows and spatio-temporally periodic solutions of classical field theories, a generalization of the Newton method to a flow in the space of loops. The feasibility of the method is demonstrated by its application to several dynamical systems, including the Kuramoto-Sivashinsky system.Comment: 14 pages, 13 figures; in N. Antoniou, ed., Proceed. of 10. Intern. Workshop on Multiparticle Production: Correlations and Fluctuations in QCD (World Scientific, Singapore 2003

Cycle expansions for intermittent diffusion

We investigate intermittent diffusion using cycle expansions, and show that a truncation based on cycle stability achieves reasonable convergence.Comment: 6 pages, revtex, 4 figure

A Perturbative Analysis of Modulated Amplitude Waves in Bose-Einstein Condensates

We apply Lindstedt's method and multiple scale perturbation theory to analyze spatio-temporal structures in nonlinear Schr\"odinger equations and thereby study the dynamics of quasi-one-dimensional Bose-Einstein condensates with mean-field interactions. We determine the dependence of the intensity of modulated amplitude waves on their wave number. We also explore the band structure of Bose-Einstein condensates in detail using Hamiltonian perturbation theory and supporting numerical simulations.Comment: 24 pages, 20 figs (numbered to 9), 6 tables, to appear in Chao

Ground Control to Niels Bohr: Exploring Outer Space with Atomic Physics

We provided an introduction to transition state theory and the connections it provides between atomic and celestial physics. We include brief discussions of historical background, recent applications in space mission design, and current research efforts.Comment: 17 pages, 3 figures, expository article, submitted to the Notices of the American Mathematical Society; better figs available at http://www.math.gatech.edu/~mason

Cycle expansions for intermittent maps

In a generic dynamical system chaos and regular motion coexist side by side, in different parts of the phase space. The border between these, where trajectories are neither unstable nor stable but of marginal stability, manifests itself through intermittency, dynamics where long periods of nearly regular motions are interrupted by irregular chaotic bursts. We discuss the Perron-Frobenius operator formalism for such systems, and show by means of a 1-dimensional intermittent map that intermittency induces branch cuts in dynamical zeta functions. Marginality leads to long-time dynamical correlations, in contrast to the exponentially fast decorrelations of purely chaotic dynamics. We apply the periodic orbit theory to quantitative characterization of the associated power-law decays.Comment: 22 pages, 5 figure

Variational method for locating invariant tori

We formulate a variational fictitious-time flow which drives an initial guess torus to a torus invariant under given dynamics. The method is general and applies in principle to continuous time flows and discrete time maps in arbitrary dimension, and to both Hamiltonian and dissipative systems.Comment: 10 page