Three-to-one internal resonances in coupled harmonic oscillators with cubic nonlinearity

Abstract

We investigate a general system of two coupled harmonic oscillators with cubic nonlinearity. Without damping, the system is Hamiltonian, with the origin as an elliptic equilibrium characterized by two distinct linear frequencies. To understand the dynamics, it is crucial to derive explicit analytic formulae for the nonlinear frequencies as functions of the physical parameters involved. In the small amplitude regime (perturbative case), we provide the first-order nonlinear correction to the linear frequencies. While this analytic expression was already derived for non-resonant cases, it is novel in the context of resonant or nearly resonant scenarios. Specifically, we focus on the 3:1 resonance, the only resonance involved in the first-order correction. Utilizing the Hamiltonian structure, we employ Perturbation Theory methods to transform the system into Birkhoff Normal Form up to order four. This involves converting the system into action-angle variables (symplectically rescaled polar coordinates), where the truncated Hamiltonian at order four depends on the actions and, due to the resonance, on one "slow" angle. By constructing suitable nonlinear and not close-to-the-identity coordinate transformations, we identify new sets of symplectic action-angle variables. In these variables, the resulting system is integrable up to higher-order terms, meaning it does not depend on the angles, and the frequencies are obtained from the derivatives of the energy with respect to the actions. This construction is highly dependent on the physical parameters, necessitating a detailed case analysis of the phase portrait, revealing up to six topologically distinct behaviors. As an application, we examine wave propagation in metamaterial honeycombs with periodically distributed nonlinear resonators, evaluating the nonlinear effects on the bandgap particularly in the presence of resonances.