PROPAGATION PROPERTIES OF BI-COUPLED NONLINEAR OSCILLATORY CHAINS: ANALYTICAL PREDICTION AND NUMERICAL VALIDATION

Abstract

Harmonic wave propagation in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. The study extends previous results obtained for the mono-coupled oscillators to bi-coupled ones. The considered bi-coupled model refers to a chain of linearly coupled mechanical oscillators characterized by on-site cubic nonlinearities in both the longitudinal and rotational degrees of freedom. Pass, stop and complex regions are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized map arising from the relevant linear stability analysis. By varying the parameters governing both the coupling between the two d.o.f. and the nonlinearity, a variable scenario of propagation regions can be obtained, which includes the limit case of mono-coupled behavior. Theoretical predictions are validated through numerical results in terms of orbits, bifurcation diagrams and basins of attraction, obtained via nonlinear map iteration. They highlight a rich variety of regular and nonregular bounded solutions.