Isochronism and tangent bifurcation of band edge modes in Hamiltonian lattices

Abstract

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.