Coarse chain recurrence, Morse graphs with finite errors, and persistence of circulations

In flow control, finite energy may be injected to push out material trapped in the attractor and to eliminate stagnation and circulate the flow. To describe such phenomena and to give a lower bound on the energy required, we generalize the existing concepts of chain recurrence. In fact, this paper introduces concepts of ``coarse chain recurrences'' and Morse graphs with finite errors. Using these concepts, we describe toy models of escape from attracting basins and elimination of stagnation by controls using finite energy, persistence of recurrent points, and singular limit behaviors where energy injections go to zero. Furthermore, we construct filtrations associated with dynamical systems, which indicate the persistence of circulations.