We analyze nonlinear aspects of the wave-particle interaction using Hamiltonian dynamics and considering a low-dimension realization of the single wave model. The wave-particle interaction plays an
important role in plasma dynamics, and the nonlinear processes resulting from this interaction are related
to the emergence of plasma instabilities and turbulence. This interaction can be represented in the (x, v)
space by regular and chaotic trajectories of particles. Regular trajectories may lead to coherent particle
acceleration while chaotic trajectories are responsible for particle heating and escape.
Often, low-dimensional approximations shed light on the dynamics of systems with many degrees of
freedom, as chaotic motion arises as one increases the number of degrees of freedom. We start with the
simple case where one particle (N = 1) is coupled to one wave (M = 1) [1]. This case is completely
integrable, so that all trajectories are regular and the nonlinear effects degenerate to particle trapping while
the wave potential pulsates. The bifurcation diagram of this simple system is already rich, with a saddlecentre coalescence and a special role of the trajectory for which the wave intensity goes through zero. On
increasing the number of particles (N = 2, M = 1), chaos arises due to the strong sensitivity in the initial
condition of relative velocity or relative position of the particles. Continuous time behaviours were also
analyzed through particle motion in the energy and wave comoving frame.
[1] J.C. Adam, G. Laval and I. Mendonça, Phys. Fluids 24, 260-267 (1981).