Generation of localization in a discrete chain with periodic boundary conditions: Numerical and analytical results

We study the generation of localization in a discrete chain composed of N subsystems and periodic boundary conditions. Strongly localized motions are studied; that is, time-periodic motions where nearly all of the energy is spatially confined to a single subsystem. For varying N, numerical results indicate that the strongly localized solutions are generated through a bifurcation from an in-phase spatially extended solution. However, in the limit as N → ∞, the bifurcation point tends to infinity, and a smooth transition from localization to nonlocalization occurs. We then present an analytic technique to complement the numerical results. It is based on the matching local asymptotic expansions of a solution branch using Fade approximants. This leads to global analytic representations of the considered solutions, valid over the entire range of the control parameter.