Response of discrete nonlinear systems with many degrees of freedom

We study the response of a large array of coupled nonlinear oscillators to parametric excitation, motivated by the growing interest in the nonlinear dynamics of microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). Using a multiscale analysis, we derive an amplitude equation that captures the slow dynamics of the coupled oscillators just above the onset of parametric oscillations. The amplitude equation that we derive here from first principles exhibits a wave-number dependent bifurcation similar in character to the behavior known to exist in fluids undergoing the Faraday wave instability. We confirm this behavior numerically and make suggestions for testing it experimentally with MEMS and NEMS resonators.

Figure 1

(Color online) Response of the oscillator array plotted as a function of reduced amplitude $g$ for three different scaled wave-number shifts determined by fixing the number of oscillators $N$: $k=0$ (for $N=100$) and $k=-0.81$ (for $N=92$), which bifurcate supercritically, and $k=1.55$ (for $N=98$) which bifurcates subcritically showing clear hysteresis. Solid and dashed lines are the positive and negative square root branches of the calculated response in (14), the latter clearly unstable. Open circles are numerical values obtained by integration of the equations of motion (1) for each $N$, with $\Delta =0.5$, ${\omega }_p=0.767445$, $ϵ\gamma =0.01$, and $\eta =0.1$. For these parameters a unity of the scaled squared amplitude ${\mid b_k\mid }^2$ corresponds to $2\times {10}^{-4}$ for the unscaled squared amplitude, and a unity of the wave-number shift $k$ corresponds to a shift of $9\times {10}^{-3}$ of the unscaled wave number. Dots show the subcritically bifurcating single-mode solution of LC [22].

Figure 2

(Color online) Stability boundaries of the single-mode solution of Eq. (12) in the $g$ vs $k$ plane. Dashed line: neutral stability boundary. Dotted line: stability boundary of the single-mode solution (13) for a continuous spectrum. Solid line: stability boundary of the single-mode solution for $N=92$ and the parameters of Fig. 1. For $k>1$ the bifurcation from zero displacement becomes subcritical and the lower stability boundary is the locus of saddle-node bifurcations (green line). Vertical and horizontal arrows mark the secondary instability transitions shown in Fig. 3.

Figure 3

(Color online) A sequence of secondary instabilities following the initial onset of single-mode oscillations in an array of 92 beams, with the parameters of Fig. 1, plotted as a function of the reduced driving amplitude $g$. Solid (dashed) lines are stable (unstable) solutions defined by (14), for $k=-0.81$, $k^{\prime }=2.90$, and $k^{″}=6.60$, corresponding to the first wave number $k$ to emerge and two shifts to $k+\Delta Q_N$ and $k+2\Delta Q_N$ respectively. Numerical integration of the equations of motion (1) for an upward sweep of $g$ (blue ×’s), followed by a downward sweep (red 엯’s) exhibits clear hysteresis and confirms the theoretical predictions for the stability of single-mode oscillations as illustrated in Fig. 2.