Coalescing Singular Values of Matrices Depending on two Parameters

and

Fundamental Matrix Solutions of Piecewise Smooth Systems

Luca Dieci

School of Mathematics, Georgia Tech

A mini course at Helsinki University of Technology,
22.04.2009

A part of the Special Year in Numerics 2008-2009
of the Finnish Mathematical Society


Lecture 1: Coalescing singular values of matrices depending on two parameters

It is well understood that there are bifurcations phenomena for systems depending on several parameters which cannot be observed, generically, by freezing one parameter at a time. One notable example is coalescing of singular values for real matrices that depend smoothly on two parameters. In this talk we review the main features of this phenomenon. In particular,
  1. we relate the existence of points where two singular values coalesce to the period doubling of the orthogonal factors of the smooth SVD along loops containing such points,
  2. we exploit this period doubling to construct numerical algorithms aimed at detecting and accurately approximating the coalescing points,
  3. we discuss how the detection of coalescing points could help identifying regions in parameters' space where certain dynamical systems exhibit hypersensitivebehavior.
[Joint work with A.Pugliese]

Lecture 2: Fundamental Matrix Solutions of Piecewise Smooth Systems

We consider the fundamental matrix solution associated to piecewise smooth differential systems of Filippov type, in which the vector field varies discontinuously as solution trajectories reach one or more surfaces. We consider the cases of transversal intersection and of sliding motion on one surface.  We also study fundamental matrices when sliding motion takes place on the intersection of two or more surfaces.
[Joint work with L.Lopez]

Lectures:

  • Wednesday 22.4.2009 at 12:15-14:00 in room U345

Contact information:
Professor Timo Eirola (Timo.Eirola@tkk.fi)
Assistant Kurt Baarman (Kurt.Baarman@tkk.fi)