Discrete Breathers in Forced Chains of Oscillators with Cubic Nonlinearities

The forced dynamics of chains of linearly coupled mechanical oscillators characterized by on-site cubic nonlinearity is investigated. The study aims to highlight the role played by the harmonic excitation on the nonlinear spatially localised dynamics of the system. Towards this goal, a map approach is employed in order to identify the chain nonlinear propagation regions under 1:1 resonance conditions. Given the latter assumption, the governing second-order difference equation refers to a perturbation of the stationary resonant response. Softening and hardening type of nonlinearities are considered and the associated unstaggered and staggered discrete breathers (DB), respectively, are discussed. Stationary DBs obtained as soliton-like solutions are identified either with sequences of the nonlinear map homoclinic primary intersection points and with an ad hoc analytic approximation; the latter is based on the idea that the nonlinearity is taken into account only in the central part of the breather whilst the tails are treated as linear excitations.