LCDS Seminars 2004

Special Joint LCDS and PDE Seminar

Monday, March 15, 2004

Lefschetz Center for Dynamical Systems Seminar

Thursday, March 25, 2004

Special Lefschetz Center for Dynamical Systems Seminar

Wednesday, April 7, 2004

Special Joint LCDS & PDE Seminar

Monday, April 12, 2004

Special Joint LCDS & PDE Seminar

Abstract :   We consider kinetic models describing two species of particles interacting via a long range repulsive potential and
a) with a reservoir at fixed temperature,
b) by collisions.
The dynamics for the first model conserves the total masses of the two species and its sharp interface limit is described by a kind of Mullins-Sekerka motion. The second dynamics models the behaviour of a binary fluid and conserves masses, momentum and energy. In the sharp interface limit in this case the velocity field satisfies the incompressible Navier-Stokes equations together with a jump boundary condition for the pressure across the interface which, in turn, moves with a velocity given by the normal component of the velocity field.

Wednesday, April 14, 2004

Special Joint LCDS & PDE Seminar

Monday, April 19, 2004

Special Lefschetz Center for Dynamical Systems Seminar

Abstract :   A new family of exact solutions is analyzed, which model 2D circulations of an ideal fluid in a uniformly rotating elliptical tank, under the semi-geostrophic approximation from meteorology and oceanography. The fluid pressure and stream function remain quadratic functions of space at each instant in time, whose fluctuations are described by a single degree of freedom Hamiltonian system depending on two conserved parameters: domain eccentricity and the constant value of potential vorticity. These parameters determine the presence or absence of periodic orbits with arbitrarily long periods, fixed points of the dynamics, and aperiodic homoclinic orbits linking hyperbolic saddle points. The energy relative to these parameters selects the frequency and direction in which isobars nutate or precess, as well as the steady circulation direction of the fluctuating flow. The canonically conjugate variables are the moment of inertia and angle of inclination of an elliptical inverse-potential-vorticity patch evolving in dual coordinates.

Monday, May 3, 2004

Lefschetz Center for Dynamical Systems Seminar

Monday, September 27, 2004

Lefschetz Center for Dynamical Systems Seminar

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Classical bifurcation theory studies dynamical systems, that depend on parameters. Frequently, trivial equilibria are assumed to exist. In an extended phase space, they form manifolds with a trivial transverse foliation. In contrast, we consider vector fields with equilibrium manifolds that are not induced by any parameters. We address the failure of normal hyperbolicity in absence of any transverse flow-invariant foliation. We call our emerging theory "bifurcation without parameters". Applications include coupled oscillators, traveling wave profiles in systems of hyperbolic balance laws, population dynamics, fluid mechanics, and many more. Motivated by several examples, we will present a variety of bifurcation scenarios. We will discuss their dynamic properties and compare them with classical transcritical, Hopf, or Bogdanov-Takens bifurcations.

Monday, October 25, 2004

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We consider spiral waves in reaction-diffusion systems. One example is the light sensitive Belousov-Zhabotinsky reaction. In the spatially homogeneous case spiral wave patterns appear, due to Euclidean $SE(2)$-symmetry, which rotate rigidly around the fixed tip position of the spiral. We are interested in the persistence of such solutions under a symmetry breaking perturbation, which still keeps a discrete lattice symmetry. In particular we study the resulting tip motions via a reduction of the original PDE-systems to vectorfields on the two-dimensional torus. Our methods include global center manifold reductions.

Monday, November 1, 2004

Lefschetz Center for Dynamical Systems Seminar

Abstract :   When can a function of several variables be written as a finite composition of functions of fewer variables? This is the question of Hilbert's problem XIII. The answer by young V.I. Arnold in the form given by Kolmogorov says: always, by functions of two variables, if we assume continuity. The answer by Vitushkin says: almost never, if we require differentiability.

In chemical engineering, we encounter a related question when we try to identify given input-output networks of coupled reactors from overall input-output data. One individual reactor model, or function, may be known ("white box") or unknown ("black box"). The hybrid model is the composition of such black- and white-box functions. Let black boxes have at most d inputs - typically much less than the total number of inputs to the network. Assuming sufficient, and at times, prohibitive differentiability, we indicate how to uniquely identify all unknown "black-box" functions in the network, from only d-dimensional data on their composition. This addresses the "curse of dimension" in data analysis, and provides extrapolability.

Results are joint work with Stefan Liebscher, Andreas Schuppert, and others.

Monday, November 8, 2004

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We ask whether a given steady state is stable or not. The three physical systems are a collisionless plasma, a semiconducting material and a surface water wave. Each system is modeled by a hyperbolic equation with or without dissipation.

Monday, November 15, 2004

Lefschetz Center for Dynamical Systems Seminar

Monday, November 22, 2004

Lefschetz Center for Dynamical Systems Seminar

Monday, November 29, 2004

Lefschetz Center for Dynamical Systems Seminar