Reducing or enhancing chaos using periodic orbits

A method to reduce or enhance chaos in Hamiltonian flows with two degrees of freedom is discussed. This method is based on finding a suitable perturbation of the system such that the stability of a set of periodic orbits changes (local bifurcations). Depending on the values of the residues, re-flecting their linear stability properties, a set of invariant tori is destroyed or created in the neighborhood of the chosen periodic orbits. An application is done on the interaction between a wave and a bunch of charged particles. This type of interaction is encountered in many branches of applied physics ranging from particle accelerators to laser physics (Free Electron Laser). Generically, this self-consistent interaction leads to an ex-ponential increase of the intensity of the wave due to the formation of clusters of charged particles. However, after a transient time, these clusters lead to undesirable oscillations of the intensity of the wave. There is therefore a crucial need to control the dynamics of these systems in order to improv