Multiple resonance in coupled Duffing oscillators and nonlinear normal modes

The resonance of $N$ linearly coupled damped Duffing oscillators with a constant frequency sinusoidal driving force acting on the first oscillator is studied analytically by calculating the fixed points of the corresponding dynamical system and numerically using a fourth-order multivariate Runge-Kutta method. For a chain with $N$ oscillators, we establish a general recursion scheme in the form of a system of equations that relates the amplitudes of the oscillators and the driving frequency, capable of describing resonance curves. We consider in detail the case of an oscillator chain with $N=2$ for high values of the driving amplitude and stiffness, and find hysteretical unstable regions in the resonance curves. In this unstable driving frequency regime, analysis of the time series reveals the presence of nonlinear normal modes visible as beating quasiperiodic oscillations.