In this project , we propose to develop Conley's theory in the context of Morse-Bott and Morse-Novikov flows, associated to Morse-Bott and Morse circular functions, respectively. More specifically, we intend to use the algebraic theory of spectral sequences together with connection matrices to study continuation properties of these flows. We want to associate the topology of the manifold to the trajectories connecting critical manifolds (singularities, respectively) of a Morse-Bott flow (Morse-Novikov flow, respectively). We will consider a Morse-Bott chain complex (Novikov chain complex, respectively) such that its differential contains information on these trajectories. We will study the connection matrices and transition matrices obtained by the sweeping method adapted to this situation, in order to understand dynamical behaviors associated with each stage of the process. The ultimate goal is to understand the bifurcations that can occur in the steps associated to transition matrices when a one-parameter family of flows is considered.
News published in Agência FAPESP Newsletter about the scholarship: