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

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.