N. Sri Namachchivaya

Evolution of Differential Forms in Dynamical Systems

Qian Wang
University of Alberta

The evolution of differential -forms facilitates the investigation of local and global properties of finite dimensional dynamical systems. The work has its origins in the 2-dimensional Poincaré-Bendixson theory for the existence of periodic orbits, the Bendixson-Dulac criteria for the non-existence of periodic orbits and the Poincaré stability condition for a periodic orbit. The main tools in extending the theory to general finite dimensional dynamical systems have been multiplicative and additive th compound matrices. The algebraic, metric and spectral properties of these matrices have provided useful insights. For example, one of the important facts is that the eigenvalues of the th additive compound matrix (or the th multiplicative compound matrix are the summation (or the multiplication) of all -tuples of eigenvalues of .

In this talk, I will present an extension of the technique developed for finite dimensional dynamical systems to a general dynamical system the form: where is a sectorial operator in a Banach space . The stability of equilibria and nonconstant periodic orbits will be discussed with applications to parabolic partial differential equations.