Applied Mathematics Seminar

On the existence and stability of rotating wave solutions to lattice dynamical systems
Friday, 21 April 2017 - 12:00 pm to 12:30 pm
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SPEAKER: Jason Bramburger (University of Ottawa)             ABSTRACT: In this talk we will investigate an infinite system of coupled lambda-omega ODEs resulting from a spatial discretization of a PDE on the infinite plane. In particular this work focuses on proving the existence and further properties of rotating wave solutions to our so-called lattice dynamical system (LDS). The work is broken down into three major components. First we demonstrate that the full lattice dynamical system can be reduced to a system of coupled phase equations, which we show to possess a rotating wave solution. Then we will examine some stability properties of this rotating wave solution by providing a fascinating link with the theory of random walks on infinite graphs. Finally we return to the full lambda-omega LDS and use the results from the coupled phase equation to extend to a rotating wave solution to the full system. This extension requires extensive results from Banach space theory and a non-standard Implicit Function Theorem to be applied in a nontrivial manner.