Multiple resonance in coupled Duffing oscillators and Nonlinear Normal Modes

We study the dynamical behavior of chains of linearly coupled Duffing oscillators, driven by a sinusoidal force acting on the first slab. General analytical solutions for resonance curves for a chain of $N$ coupled Duffing oscillators are presented and discussed. For a system of two coupled oscillators, analytical resonance curves are completely depicted for high values of stiffness ($\gamma > 20$). Furthermore, resonant nonlinear normal modes (NNM) of quasi-periodic oscillations, are found within hysteresis regions, and theoretically characterized through a suitable closed expression.