Dynamics of relative phases: Generalised multibreathers

For small Hamiltonian perturbation of a Hamiltonian system of arbitrary number of degrees of freedom with a normally non-degenerate submanifold of periodic orbits we construct a nearby submanifold and an `effective Hamiltonian' on it such that the difference between the two Hamiltonian vector fields is small. The effective Hamiltonian is independent of one coordinate, the `overall phase', and hence the corresponding action is preserved. Unlike standard averaging approaches, critical points of our effective Hamiltonian subject to given action correspond to exact periodic solutions. We prove there has to be at least a certain number of these critical points given by global topological principles. The linearisation of the effective Hamiltonian about critical points is proved to give the linearised dynamics for the full system to leading order in the perturbation. Hence in the case of distinct eigenvalues which move at non-zero speed with epsilon, the linear stability type of the periodic orbit can be read off from the effective Hamiltonian. Our principal application is to networks of oscillators or rotors where many such submanifolds of periodic orbits occur at the uncoupled limit - simply excite a number N greater than or equal to 2 of the units in rational frequency ratio and put the others on equilibria, subject to a non-resonance condition. The resulting exact periodic solutions for weak coupling are known as multibreathers. We call the approximate solutions given by the effective Hamiltonian dynamics, 'generalised multibreathers'. They correspond to solutions which look periodic on a short time scale but the relative phases of the excited units may evolve slowly. Extensions are sketched to travelling breathers and energy exchange between degrees of freedom.