Stabilizing a wave amplified by a beam of particles with test-waves R. Bachelard ∗ , C. Chandre ∗ , D. Fanelli † , X. Leoncini ∗ , M. Vittot ∗ Abstract The intensity of an electromagnetic wave interacting self-consistently with a beam of charged particles as in a free electron laser, displays large oscillations due to an aggregate of particles, called the macro-particle. In this article, we propose a strategy to stabilize the intensity by destabilizing the macro-particle. This strategy involves the study of the linear stability (using the residue method) of a specific periodic orbit of a mean-field model. As a control parameter - the amplitude of an external wave - is varied, a bifurcation occur in the system which have drastic effect on the modification of the self-consistent dynamics, and in particular, of the macro-particle. We show how to obtain an appropriate tuning of the control parameter which is able to strongly decrease the oscillations of the intensity without reducing its mean-value. Extensive abstract The self-consistent interaction between an electromagnetic wave and a beam of charged particles is ubiquitous in many branches of physics, e.g. accelerator and plasma physics. For instance, it plays a crucial role in the Free Electron Laser, which is used to generate a tunable, coherent, high power radiations. Such devices differ from conventional lasers in using a relativistic electron beam as its lasing medium. The physical mechanism responsible for the light emission and amplification is the interaction between the beam and a wave, which occurs in presence of a magnetostatic periodic field generated in an undulator. Due to the effect of the magnetic field, the electrons are forced to follow sinusoidal trajectories, thus emitting synchrotron radiation. This initial seed, termed spontaneous emission, acts as a trap for the electrons which in turn amplify it by emitting coherently, until the laser effect is reached. The coupled evolution of radiation field and N particles can be modeled within the frame- work of a simplified Hamiltonian picture [1]. The N +1 degree of freedom Hamiltonian displays a kinetic contribution, associated with the particles, and a potential term accounting for the self-consistent coupling between the particles and the field. Hence, direct inter-particle inter- actions are neglected, even though an effective coupling is indirectly provided because of the interaction with the wave. The linear theory predicts [1], for the amplitude of the radiation field, a linear exponential instability and a late oscillating saturation. Inspection of the asymptotic phase-space suggests that a bunch of particles gets trapped in the resonance and forms a clump that evolves as a single macro-particle localized in phase space. The untrapped particles are almost uniformly distributed between two oscillating boundaries, and populate the so-called chaotic sea. Furthermore, the macro-particle rotates around a well defined center and this peculiar dynamics is shown to be responsible for the macroscopic oscillations observed for the intensity [2, 3]. It can be therefore hypothesized that a significant reduction in the intensity fluctuations can be gained by implementing a dedicated control strategy, aimed at destroying the macro- particle in space. As a side remark, note that the size of the macro-particle is directly related to the bunching parameter, a quantity of paramount importance in FEL context[3]. For example, a static electric field [4] can be used to increase the average wave power. While the chaotic particles are simply accelerated by the external field, the trapped ones transmit the extra energy to the radiation field, thus being responsible for the amplification of the latter. Furthermore, the experiment by Dimonte and Malmberg [5] suggests that a strategy based on the destruction of the macro-particle may reduce the oscillations of the intensity of the wave. The dynamics can be investigated from a topological point of view, by looking at the phase space structures. In the framework of a simplified mean field description, i.e. the so-called test-particle picture where the particles passively interact with a given electromagnetic wave, ∗ Centre de Physique Théorique, CNRS Luminy, Case 907, F-13288 Marseille Cedex 9, France † Theoretical Physics Group, School of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, U.K.