Isochronism and tangent bifurcation of band edge modes in Hamiltonian lattices

In Physica D [S. Flach, Physica D 91 (1996) 223], results were obtained regarding the tangent bifurcation of the band edge modes (q=0,π) of nonlinear Hamiltonian lattices made of N coupled identical oscillators. Introducing the concept of partial isochronism which characterises the way the frequency of a mode, ω, depends on its energy, ε, we generalize these results and show how the bifurcation energies of these modes are intimately connected to their degree of isochronism. In particular, we prove that in a lattice of coupled purely isochronous oscillators (oscillators with an energy-independent frequency), the in-phase mode (q=0) never undergoes a tangent bifurcation whereas the out-of-phase mode (q=π) does, provided the strength of the nonlinearity in the coupling is sufficient. We derive a discrete nonlinear Schrödinger equation governing the slow modulations of small-amplitude band edge modes and show that its nonlinear exponent is proportional to the degree of isochronism of the corresponding orbits. This equation may be seen as a link between the tangent bifurcation of band edge modes and the possible emergence of localized modes such as discrete breathers.