Spectral Analysis, Stability and Bifurcation in Modern Nonlinear Physical Systems (12w5073)

Arriving in Banff, Alberta Sunday, November 4 and departing Friday November 9, 2012


Paul Binding (University of Calgary)

(University of Surrey)

(Kyushu University)

Igor Hoveijn (Noordelijke Hogeschool Leeuwarden)

(Helmholtz-Zentrum Dresden-Rossendorf)

(McMaster University)


Linearised stability analysis of stationary and periodic solutions of both finite and infinite dimensional dynamical systems is a central issue in many (physical) applications. Such systems usually depend on parameters, so an important question is what happens to stability when the parameters are varied. This implies that one has to study the spectrum of a linear operator and its dependence on parameters. Moreover, systems arising in physics and other applications often possess special structure, for example Hamiltonian systems. Therefore spectrum and Jordan structure no longer suffice to characterize equivalent systems (under smooth coordinate transformations) but additional invariants are needed. Identifying and interpreting these in infinite dimensional systems seems more involved than in finite dimensional situations. For example, one may consider the symplectic or Krein signature for imaginary eigenvalues in linear finite dimensional Hamiltonian systems. When such eigenvalues meet as the parameters vary, the existence of additional invariants causes non-generic behaviour. In particular, a collision of eigenvalues on the imaginary axis may have dynamical consequences since the stability may change, depending on the additional invariants. At such a collision the boundary of the so called stability domain in parameter space may have singularities. This phenomenon occurs in numerous applications and it may have various physical consequences and interpretations. On the other hand stability questions can also be studied by index theory (Morse index, Maslov index). These approaches are not unrelated; for example the symplectic or Krein signature is connected to the Morse index.

The aim of the meeting is to bring together people from different disciplines each studying these stability questions from their own perspectives, rooted in established traditions. Recent mathematical and physical facts point out the existence of hidden new intriguing connections between the original researches of the invitees. We feel that an atmosphere of co-education and co-discovery of the BIRS workshop will help us to clarify the existing analogies, stimulate active cross-usage of methods and approaches that finally will result in establishing new links connecting different mathematical disciplines and seemingly different applications. One of the central aims is to give a deeper and broader understanding to all participants of stability questions and to unify various methods and treatments which may in turn lead to the birth of new directions of investigation.

It is expected that participants of the workshop will establish spontaneous contacts with each other according to their tastes and interests. However, selection of some critical directions of research seems to be necessary to focus the efforts. Ideally, the research topics should be formulated in such a manner that both applied and pure scientist feel comfortable because of definite mutual interest. Taking this into account we think that the possible focus groups will be:

1) Instability index, Maslov index (geometry, topology etc), geometrical phase, geometrical optics
2) Instabilities in PDEs from elasticity, fluids, magnetohydrodynamics, fluid-structure interactions etc.
3) Perturbation and asymptotic analysis of multiparameter eigenvalue problems (eigencurves, singularities of different surfaces related to stability)
4) Hamiltonian and reversible systems, dissipation-induced instabilities (development of a general theory and interpretations for applications)
5) Numerical analysis, applied linear algebra (computational aspects related to tracking the eigenvalues and eigenvectors, parallel transport of eigenvectors, computation of indices, finding multiple eigenvalues of non-Hermitian operators, pseudospectra)
6) Symmetries and stability (equivariance, fundamental symmetry, influence of symmetries hidden in the boundary conditions)