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.

Figure 1

Appearance of hysteresis. Resonance curves of the first oscillator with amplitude $Q_1$ and fixed parameters ${\omega }_0^2=1$, $\zeta =0.1$, $\delta =1$. (a) Varying $\gamma$ and $\mathrm{\Omega }$ at fixed $F=0.1$. (b) Varying $F$ and $\mathrm{\Omega }$ at fixed $\gamma =0.1$.

Figure 2

Appearance of hysteresis. Resonance curves of the second oscillator with amplitude $Q_2$ and fixed parameters ${\omega }_0^2=1$, $\zeta =0.1$, $\delta =1$. (a) Varying $\gamma$ and $\mathrm{\Omega }$ at fixed $F=0.1$. (b) Varying $F$ and $\mathrm{\Omega }$ at fixed $\gamma =0.1$.

Figure 3

Beat response in the system's time series, and clusters of frequencies in Fourier space. Time series (left) and Fourier transform (right). Parameters: ${\omega }_0^2=1$, $\zeta =0.1$, $\delta =1$, $F=0.1$; initial conditions: $x_{1,2}\left(0\right)=0.3$, ${\dot{x}}_{1,2}\left(0\right)=0$. Frequencies: $\mathrm{\Omega }=1.97(\gamma =20),\mathrm{\Omega }=2(\gamma =30),\mathrm{\Omega }=2.1(\gamma =50,60)$ from top to bottom row. The line lin-log (see text) has been scaled by a 2 factor to emphasize the correspondence between slopes.

Figure 4

Multiple clusters of frequencies in Fourier space. Detailed Fourier spectrum of time series shown for $\gamma =30$ over a wider range of frequencies $\omega$ (c.f. Fig. 3). A main cluster of frequencies is identified for $\omega \sim 2$, and two secondary clusters as $\omega$ increases around $\omega \sim 6$ and $\omega \sim 10$. The amplitude of each cluster reduces about an order of magnitude as $\omega$ increases from cluster to cluster.

Figure 5

Approximation of beating NNMs. Theoretical time series (left) and Fourier transform (right) obtained from Eq. (11). Parameters: $\mathcal{N}=2$, $\nu =0.02$, $a=10$ in black and $\mathcal{N}=12$, $\nu =0.0125$, $a=8$ in red.

Figure 6

Quasiperiodic trajectories folded tori. 3D projections of the phase-space coordinates for the same parameters as the time series shown in Fig. 3 for $\gamma =30$, (a) including the velocity of the first oscillator (${\dot{x}}_1$), and (b) including the velocity of the second oscillator (${\dot{x}}_2$).

Figure 7

Poincaré section displaying closed curves. Poincaré map corresponding to results in Fig. 3 for $\gamma =30$. These self-intersecting patterns represent the folding of the tori in phase space.

Figure 8

Configuration space, 2D projection of any of the folded tori in Fig. 6. The strips correspond to Poincaré sections for the same variables taken with different initial points.

Figure 9

Theoretical resonance curves for both oscillators' amplitudes ($A_i$), comparing the NR (markers) and eigenvalue decomposition (solid line) methods. The first oscillator is on top. Parameters used: ${\omega }_0^2=1$, $\zeta =0.1$, $\delta =1$, $\gamma =20$ and $F=0.1$. The forward and backward labels refer to applying the NR method increasing and decreasing $\mathrm{\Omega }$ respectively.

Figure 10

Comparison between the numerical and theoretical resonance curves for both oscillators' scaled amplitudes $Q_i$. The first oscillator is on top. Parameters used: ${\omega }_0^2=1$, $\zeta =0.1$, $\delta =1$, $\gamma =30$ and $F=0.1$. The forward and backward labels refer to applying the RK4 method increasing and decreasing $\mathrm{\Omega }$ respectively.

Figure 11

Phase space ($x_1,{\dot{x}}_1$) of the first oscillator related to Fig. 3 in the main manuscript for $\gamma =30$.