Three-to-one internal resonances between acoustic and optical waves in metamaterials with nonlinear resonators

We investigate the modal interactions between two generalized oscillators representing the acoustic and optical waves obtained as solutions of the wave propagation equations across a metamaterial conceived as cellular hosting 2D structure augmented by intracellular resonators. Without damping, the system is Hamiltonian, with the origin as an elliptic equilibrium characterized by two distinct linear frequencies. To understand the underlying dynamics, it is crucial to derive explicit analytical formulae for the nonlinear frequencies as functions of the physical parameters. 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, the interest is here placed on wave interactions in the context of resonant or nearly resonant scenarios. In particular, we focus on 3:1 internal resonance, the only resonance involved in the first-order correction. We then address the challenging strongly resonant case in which the detuning is small with respect to the perturbative parameter. Unlike standard approaches, here we capture the full complexity of resonant dynamics, revealing a richer, more intricate topological features. Utilizing the Hamiltonian structure, we employ Perturbation Theory to transform the system into Birkhoff Normal Form up to order four. This involves converting the system into action–angle variables, 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, which reveals up to six topologically distinct behaviors. In each instance, we describe the nonlinear normal modes (elliptic/hyperbolic periodic orbits, invariant tori) and their stable and unstable manifolds of the truncated Hamiltonian. For the computations, we examine resonant wave interactions in metamaterial honeycombs with periodically distributed Duffing-type resonators, specifically addressing the nonlinear effects on the bandgap. More precisely, in view of the metamaterial design, our analysis allows one to identify the values of the modal mass and stiffness of the resonators, that maximize the beneficial effect of nonlinearity in enlarging the bandgap width.