Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators

Abstract

We address the issue of spatially localized periodic oscillations in coupled networks - so-called discrete breathers - in a general context. This context is concerned with general conditions which allow continuation of periodic solutions of vector fields. One advantage of our approach is to encompass in the same mathematical framework the cases of conservative and dissipative systems. An essential feature is that we allow the period to vary. In particular, we deduce existence of discrete breathers in networks where each site has an equilibrium and some sites have a limit cycle, and in Hamiltonian networks without requiring local anharmonicity. The latter case is dealt with by considering the persistence of families of periodic solutions in the more general context of systems with an integral, not just Hamiltonian ones.