N. Sri Namachchivaya
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Evolution of Differential Forms in Dynamical Systems

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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 PoincarĂ©-Bendixson theory for the existence of
periodic orbits, the Bendixson-Dulac criteria for the non-existence of periodic
orbits and the PoincarĂ© 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.