Resonance phenomena in dynamical systems and control

Dmitri Vainchtein, Mechanical and Environmental Engineering, University of California, Santa Barbara

We discuss the behavior of systems for which the resonance phenomena are an intrinsic part of internal dynamics and also certain applications of the idea of resonances in control theory.

In the first part of my talk we discuss a couple of systems where the resonance phenomena is a part of internal dynamics. We show how the multiple separatrix crossings lead to the chaotic advection and mixing in a liquid drop and how capture into the resonance and scattering on a resonance result in chaotization of the motion of charged particles in electromagnetic fields.

In the second part we discuss the application of resonance methods to control. Using vortex dynamics as an example, we show how the understanding of the internal dynamics of a system leads to constructing an optimal control. We start with the problem of controlling of a system of two interacting vortex patches. We compare two possible approaches. The first one is to use the method of flat coordinates. The other one is to apply the averaging technique using control field that is a small perturbation compared with the original velocity field. The second problem is controlling the separation of a co-rotating pair of point vortices a souce/sink field. We use Pontryagin's maximum principle to prove that the optimality is achieved when the control is kept in phase with the internal rotation of vortices.