LCDS Seminars 2007

Monday, January 22, 2007

Special PDE & Lefschetz Center for Dynamical Systems Seminar

Monday, March 12, 2007

Joint LCDS and PDE Seminar

Monday, March 19, 2007

Lefschetz Center for Dynamical Systems Seminar

Abstract:   We consider two special reducible cases for the Volterra and the Levin-Nohel retarded equations with infinite delay. We address global questions like the existence of Liapunov functions and, consequently, of attractors for the nonlinear systems generated by these equations as well as the attractors for the reduced systems. Some Morse-Smale structures of these two reducible equations are also identified.

Monday, October 8, 2007

Lefschetz Center for Dynamical Systems Seminar

Abstract:   I will start with the deterministic dynamics generated by a vector field that has several unstable critical points connected by heteroclinic orbits thus forming a heteroclinic network. A perturbation of this system by white noise will be considered. I will study the limit of the resulting stochastic system in distribution (under appropriate time rescaling) as the noise intensity vanishes. It is possible to describe the limiting process in detail, and, in particular, I will give conditions guaranteeing that it is Markov or non-Markov. This asymptotic result implies interesting corrections for the exit asymptotics provided by the classical wentzell-Freidlin theory.

Monday, October 15, 2007

Lefschetz Center for Dynamical Systems Seminar

Abstract:   Fluids undergo transition to turbulence when slight perturbations cause the laminar solution to become turbulent. Historically, efforts to understand transition to turbulence have focused on a neighborhood of the laminar solution. This talk will describe a more geometric approach based on steady solutions and traveling waves that act as intermediaries between the laminar solution and turbulence. Another theme is to demonstrate the relevance of recurrences and accurate computations of those recurrences to the motion of turbulent fluids.

Monday, October 22, 2007

Lefschetz Center for Dynamical Systems Seminar

Abstract:   Spiral waves are planar patterns that arise in many natural systems. Part of their fascination is due to the intriguing instabilities, such as meandering and drifting, core and far-field breakup, and spatio-temporal period doubling, that they exhibit. Among the challenges for theoretical studies of spirals is the task of relating these instabilities to spiral spectra. In this talk, I will present recent results that predict certain aspects of spiral spectra and outline how these can be used to explain some of the observed instabilities of spiral waves.

Monday, October 29, 2007

Lefschetz Center for Dynamical Systems Seminar

Abstract:   Generally speaking animals do not waste too much of any scarce resource, of which energy is one. Similarly, if robots are ever to be practical, reasonable energy efficiency would be useful. In loose terms, we have been trying to understand legged locomotion from an energetic point of view. One approach is to make energy- efficient legged machines, the most striking of which are those with no motors that only walk down gentle slopes. Another approach is to study energy-use of humans. Finally one can make mathematical analyses of simple mechanical models. One idea that comes out of such is this: perhaps the non-holonomic nature of the intermittent contact in walking contributes to the stability of walking.

Monday, November 5, 2007

Lefschetz Center for Dynamical Systems Seminar

Abstract:   Introducing a small hole into the phase space of an ergodic dynamical system causes almost every trajectory to eventually escape. Despite this, such open systems may exhibit interesting dynamics and admit physically relevant invariant and conditionally invariant measures. We give a brief introduction to the central questions involved in the study of open systems and present recent results regarding dispersing billiards with holes. This is joint work with Lai-Sang Young and Paul Wright.

Monday, November 12, 2007

Lefschetz Center for Dynamical Systems Seminar

Abstract:   We present some large deviation results for ergodic averages for dynamical systems which have some hyperbolicity but are not necessary uniformly hyperbolic. Examples are the Lorentz gas, some quadratic maps of the interval, Henon maps, etc. We also discuss some applications to nonequilibrium statistical mechanics. This is a joint work with L.S. Young.

Monday, November 26, 2007

Lefschetz Center for Dynamical Systems Seminar

Abstract:   Techniques inherited from dynamical systems theory can be applied to derive asymptotic properties for Navier-Stokes flows in 2D exterior domains. In both the stationary and time-periodic case, interpreting the coordinate parallel to the limiting (nonzero) velocity at infinity as a "time coordinate" allows one to write the 2D Navier-Stokes equations in a form which is reminiscent of a nonlinear parabolic equation in 1D, with the somewhat surprising result that both type of flows cannot be distinguished from one another far in the downstream direction.

Monday, December 10, 2007

*** Special LCDS & PDE Seminar ***

Lefschetz Center for Dynamical Systems Seminar

Abstract:   The following scenario has been seen in many non-integrable, dispersive, nonlinear PDE over the last 25 years: two solitary waves are propagated on a collision course. Above some critical velocity vc, they simply bounce off each other. Below vc they may be captured and merge into a single localized mass, or they may interact a finite number of times before escaping each other's embrace. Whether they are captured, and how many times the solitary waves interact before escape, depends on the initial velocity in a complicated manner, often remarked, though never previously shown, to be a fractal (a chaotic scattering process). This has been observed in coupled NLS, sine-Gordon, phi-4, and others. These PDE systems are commonly studied by (nonrigorously) deriving a reduced set of ODE that numerically reproduce this behavior. Using matched asymptotics and Melnikov integrals, we give asymptotic formulas for the critical and for certain salient features of the fractal structure. We derive a discrete-time iterated map through which the entire structure can be unravelled. Surprising connections are made to other well-known dynamical systems.