PDE/Analysis Seminar - Andres Zuniga - Energy-prescribed connecting orbits for gradient systems



Speaker: Andres Zuniga (McMaster University)

Title: Energy-prescribed connecting orbits for gradient systems.

Abstract: We study solutions of second-order gradient systems of ODEs for general potentials (time independent Hamiltonian systems),in arbitrary dimensions. Given a real number c, we address potentials V whose c-sublevel set splits as the disjoint union of closed sets Vc- and Vc+. Under this assumption, and by means of an energy-constrained variational method, we obtain the existence of bounded solutions q_c to the ODE system with mechanical energy equal to -c whose trajectories connect Vc- and Vc+. The connecting orbits are classified into brake type, heteroclinic type or homoclinic type,depending on the topology of the level sets. Next, we illustrate applications to the case of double-well potentials, and for potentials associated to systems of duffing-type and of multiple-pendulum-type. In each case, we prove a convergence result of the family of solutions q_c, as the parameter c tends to a critical value of V. This work is a collaboration with Francesca Alessio (UNIVPM) and Piero Montecchiari (UNIVPM).

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