Nonlinear Dynamical Band Structure Induced by Frequency-Dependent Nonlinearity in Duffing Oscillators

Abstract

We demonstrate that a nonlinear medium whose nonlinear susceptibility is explicitly frequency-dependent such as graphene-exhibits nonlinear dispersive dynamical states governed by the Duffing equation. These states selectively suppress or support certain frequency components of the driving field, forming a nonlinear band structure without any spatial periodicity. This reveals a new mechanism of nonlinear frequency filtering originating solely from the dynamical response of the nonlinear oscillator. The nonlinear dispersive medium
modifies the effective Floquet quasi-energy landscape of the driven Duffing oscillator. The Jacobi elliptic solutions exist only in regions where the nonlinear Floquet modes are stable, forming frequency-selective “nonlinear Floquet bands”. Outside these bands the Floquet multipliers diverge, resulting in forbidden frequencies.