LCDS Seminars 2000

Monday, March 6, 2000

** Lefschetz Center for Dynamical Systems Seminar**
See Thursday, March 9 LCDS Seminar

The 2000 J.P. LaSalle Memorial Lecture

Lefschetz Center for Dynamical Systems Seminar

Thursday, March 9, 2000

Lefschetz Center for Dynamical Systems Seminar
The 2000 J. P. LaSalle Memorial Lecture

Monday, March 13, 2000

Lefschetz Center for Dynamical Systems Seminar

•  Abstract :   What semiconductor laser theory, fiber optics, surface water waves and acoustic waves have in common? Although these systems are seemingly disconnected and have quite different physical nature, they can be viewed as complex systems composed out of interacting particles or waves. There is a general theoretical framework for their statistical description, called weak turbulence theory. One can obtain a closed equation describing the time evolution of such systems, called kinetic equation. I will explain what classes of stationary solutions kinetic equation has, and how understanding of surface water waves can lead to better design of semiconductor lasers.

Monday, March 20, 2000

Lefschetz Center for Dynamical Systems Seminar

Abstract :   A Melnikov theory is developed for 2-D heteroclinic manifolds in a 3-D geometry, for the case where the perturbation is not necessarily differentiable in the relevant small parameter. The motivation for this theory stems from fluid mechanics, in which a Navier-Stokes perturbation of a Euler flow need not possess smoothness
in the viscous parameter. The theory is applied in such a setting, to address the manifold splitting due to small viscosity. Some preliminary work in analysing the (experimentally observed) bubble-vortex breakdown phenomena using this approach is presented.

This research is in collaboration with Igor Mezic ( Harvard University ) and Chris Jones ( Brown University ).

Monday, April 3, 2000

Lefschetz Center for Dynamical Systems Seminar

Tuesday, April 4, 2000

Joint Seminar - Lefschetz Center for Dynamical Systems and Center for Fluid Mechanics

Abstract :   The accuracy of drifter paths calculated from the surface velocity field from the Princeton Ocean Model of the Gulf of Mexico is assessed with observed drifter data. The approach is to project the model results onto geometric orthogonal functions. The projection is further constrained to exactly match the observed drifter velocities. In a Eularian reference the model velocity field and the reconstructed field are quite similar. In a Lagrangian reference, however, the paths integrated from the two fields differ appreciably. This difference is quantified by two Lagrangian metrics. The Lagrangian metrics for the constrained projections are about an order of magnitude better than the unconstrained projections. Some speculations on assimilation of Lagrangian data directly into predictive models are offered.

Monday, April 17, 2000

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We present a description of optical pulse propagation in fibers with randomly varying values of chromatic dispersion. The corresponding mathematical model is the nonlinear Schroedinger equation with a randomly varying dispersion coefficient. Pulse propagation is described by a statistical description of pulse degradation during its propagation along the fiber. Using path integrals along trajectories, we obtain the Fokker-Planck equation for the probability distribution function. We will present different statistical scenarios of pulse evolution in terms of the distribution function which are of practical interest.

Wednesday, May 24, 2000

Lefschetz Center for Dynamical Systems Seminar

Abstract :   I will present the derivation and analysis of the anisotropic averaged Euler equations for incompressible hydrodynamics. This model is obtained by "fuzzying" the Lagrangian flow map of the Euler equations over spatial scales smaller than some number alpha, and averaging on the volume-preserving diffeomorphism group. The resulting system of equations evolves the pair $(u,F)$, where $ u $ the macroscopic velocity field and $ F $ is a symmetric fluctuation tensor measuring the deviation of the flow from the mean. After solving this system, one may then "correct" the macroscopic velocity to order $ alpha^{2} $. Well-posedness results will be given, and the channel flow approximation will be discussed.

Monday, June 12, 2000

Lefschetz Center for Dynamical Systems Seminar

Lefschetz Center for Dynamical Systems Seminar

Abstract :   For systems described by ordinary differential equations, we consider the problem of how to design a feedback law to alter a subcritical steady-state/Hopf bifurcation to a supercritical one, when the bifurcating mode is linearly unstabilizable. Under certain nondegeneracy conditions, algebraic necessary and sufficient conditions of stabilizability are given. The effect of magnitude saturation of feedback controllers are analyzed. Geometric interpretations of stabilizability are given for some of the conditions. And sufficient conditions are obtained for systems governed by functional differential equations.

In models describing rotating stall in gas turbine engines, subcritical Hopf bifurcations are asssociated with abrupt inception of instabilities and hysteresis. We give a qualitative analysis on the effects of magnitude saturation, dandwidth, and rate limits on active control of rotating stall. The theoretical analysis is compared with experiments n a low speed, single stage compressor.

Monday, September 25, 2000

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We model a dilute gas Bose--Einstein condensate trapped in a standing light wave by the cubic nonlinear Schr\"odinger equation with an elliptic function potential. New families of stationary solutions are presented and their stability is examined using analytic and numerical methods. Jacobi elliptic $Dn(x, k)$ solutions are found to be stable for defocusing, whereas Jacobi elliptic $Cn(x, k)$ solutions are found to be stable for focusing. The linearized stability calculations allow us to generate a set of criteria concerning the stability and instability of the various families of solutions. Our results imply that for defocusing (repulsive BEC), a large number of condensed atoms is sufficient to form a stable, periodic condensate. For

Monday, October 16, 2000  

Lefschetz Center for Dynamical Systems Seminar

•  Abstract :   The Skyrme model (1961) was one of the first attempts to describe elementary particles as localized in space solutions of nonlinear PDEs. The fields take their values in $SU(2)=S^3$ and stabilize at spatial infinity. Thus, the configuration space splits into different sectors (homotopy classes) with a constant integer topological charge (the degree) in each sector. Faddeev's model (1975) was designed to provide additional internal structure (knottedness) to the localized solutions. The fields take their values in the two-dimensional sphere and the topological charge is the Hopf invariant. I will discuss some old and new results for these models.

Monday, October 23, 2000

Joint Seminar, Lefschetz Center for Dynamical Systems and The Center for Fluid Mechanics

Abstract :   We consider the statistics of pairs of subsurface, free-drifting floats in the North Atlantic . Previous observations from the atmosphere and the ocean surface are discussed, as is the theoretical framework (turbulence) in which those observations have been interpreted. The present results are likewise compared to predictions from 2-D turbulence theory, and some agreement is found; however, alternate dynamical paradigms may be consistent as well.

Monday, November 6, 2000

Special LCDS/PDE Seminar

Special Lefschetz Center for Dynamical Systems Seminar

Joint Seminar, Lefschetz Center for Dynamical Systems and
The Center for Fluid Mechanics Seminar

Abstract :   Electrospinning is a process which creates sub micron sized fibers from fluid forced through a millimeter sized nozzle by an electric field. We present evidence from theory and experiments that the essential mechanism underlying this process is a rapidly whipping fluid jet. An asymptotic approximation of the equations of electrohydrodynamics is developed so that quantitative comparisons with experiments can be carried out. The onset threshold for electrospinning is demonstrated to quantitatively agree with the onset of a whipping instability. Scaling laws for the whipping frequency are derived. It is demonstrated that quantitative features of the instability depend sensitively on the shape and material properties of the nozzle.

Monday, December 4, 2000

Lefschetz Center for Dynamical Systems Seminar

Abstract :   It is observed that in the presence of small-scale eddies the transport of large-scale vector quantities is accompanied with depleted, and in some cases even "negative", diffusion. This phenomenon is the result of the weak nonlinear interaction between large and small scales in incompressible fluid. This interaction is described by the eddy viscosity, a four-tensor that relates the large-scale deviatoric stress to the large-scale rate of strain. The goal of this talk is to analyze the eddy viscosity for large Reynolds numbers by means of Homogenization of the Navier-Stokes equations in the case of two-dimensional cellular flows. In this talk I plan to discuss some new analytical and numerical results for eddy viscosity, such as the saddle-point variational principles for a class of non-symmetric, nonlocal operators and the dynamical systems approach to the weak convergence of solutions for Hadamard ill-posed equations.