Pattern selection in parametrically driven arrays of nonlinear resonators

We study the problem of pattern selection in an array of parametrically driven nonlinear resonators with application to microelectromechanical and nanoelectromechanical systems, using an amplitude equation recently derived by Bromberg, Cross, and Lifshitz [Phys. Rev. E 73, 016214 (2006)]. We describe the transitions between standing-wave patterns of different wave numbers as the drive amplitude is varied either quasistatically, abruptly, or as a linear ramp in time. We find interesting hysteretic effects, which are confirmed by numerical integration of the original equations of motion of the interacting nonlinear resonators.

Figure 1

(Color online) Stability boundaries of the single-mode solution (7) of the BCL amplitude equation (6) in the $g$ vs $k$ plane. Dashed line: neutral stability curve $g=k^2$. Dotted line: stability boundary of the single-mode solution (7) for a continuous spectrum $(Q\to 0)$. Solid lines: stability boundary of the single-mode solution for $N=92$ and the parameters $\Delta =0.5$, $q_p=73\pi ∕101$, and $ϵ=\delta =0.01$ (giving $k_0\simeq -0.81$ and $\Delta Q_N\simeq 3.70$). Upper boundary (solid black curve): the value of $g$ for which the eigenvalue $\lambda \left(\Delta Q_N\right)$ turns positive. Lower boundary for $k<1$ [solid red (dark gray) curve]: the neutral stability curve $g=k^2$. Lower boundary for $k>1$ [solid green (light gray) line]: the locus of saddle-node bifurcations $g=2k-1$. Vertical and horizontal arrows mark the secondary instability transitions shown in Fig. 2 and discussed in Sec. 4; the blue upward and right-pointing arrows are for upward sweeps and the red downward and left-pointing arrow, for downward sweeps.

Figure 2

(Color online) The amplitude of single-mode oscillations as a function of the reduced drive amplitude $g$. The parameters are the same as in Fig. 1. Solid lines show the analytical values (10) of the amplitudes $b_k^2$ of the modes $k_n$ (for $n=0,\dots ,3$). Note that the $k_0$ mode bifurcates supercritically, whereas all the other modes start at a saddle-node bifurcations. All modes terminate at the values of $g$ for which they become Eckhaus unstable. Symbols show numerical calculations—where in the online version blue is used for upward sweeps of $g$ and red is used for downward sweeps—as follows: (a) +’s and ◻’s show upward and downward sweeps, respectively, of the original equations of motion (1) of the coupled resonators. (b) △’s and ▽’s show upward and downward sweeps, respectively, of the BCL amplitude equation (6). (c) ∗’s and ○’s show upward and downward sweeps, respectively, of the truncated mode expansion equations (16) for the seven modes $b_{-3}$ to $b_3$.

Figure 3

(Color online) Time evolution of the amplitudes of the four largest modes that participate in the Eckhaus transition from the initial $k_0$ pattern to the $k_1$ pattern, obtained by a numerical integration of the seven truncated mode equations (16), for modes $b_{-3}–b_3$, using the same parameters as in Fig. 1. The value of the control parameter is changed from $g=10$ to $g=11$ at $\tau =0$, causing the initial $k_0$ pattern to become unstable. The decay of the amplitude $b_0$ is followed by the rise of $b_1$ to its expected steady-state value (10), where it is clearly seen that during the transition other modes—including the unstable $k_{-1}$ mode—have a nonzero amplitude.

Figure 4

(Color online) Time evolution of the amplitudes of the four largest modes as the control parameter is changed from $g=20$ to $g=19$, obtained by a numerical integration of the seven truncated mode equations (16), for the amplitudes $b_{-3}–b_3$, using the same parameters as in Fig. 1. As the value of $g$ drops below the saddle node value of the $k_3$ wave number, its amplitude drops abruptly to zero. Then, the smallest possible wave number which has the largest linear growth rate over the zero solution, $k_0$, grows to reach its steady-state value (10). Nevertheless, after a short transient the $k_0$ pattern decays through an Eckhaus instability and the $k_1$ pattern grows to its steady-state value.

Figure 5

(Color online) The linear growth rate ${\lambda }_{g,k_0}\left(Q\right)$ as a function of the wave number shift $Q$, plotted for different values of the control parameter $g$. The horizontal axis labels $Q$ in units of $\Delta Q_N$, which for a finite system of $N=500$ resonators, with $k_0\simeq -0.075$, has the value of $\Delta Q_N\simeq 0.69$ (all other parameters are the same as in Fig. 1). The Eckhaus instability of the initial $k_0$ pattern occurs for these parameters at $g\simeq 5.13$.

Figure 6

(Color online) The wave number shift $Q_{\max }$ with the maximal growth rate (in units of $\Delta Q_N$) as a function of $g$. The blue solid line shows $Q_{\max }$ for the same parameters used in Fig. 5, and the green dashed line gives $Q_{\max }$ for an infinite system (17). The open red circles are the wave number shifts that are observed numerically by integrating the BCL amplitude equation (6). The full black circles are wave number shifts that are obtained by a numerical integration of the original equations of motion (1), calculated for select values of $g$. The numerical calculations are initialized with the $k_0$ solution and random small-amplitude noise, to initiate the growth of competing patterns. We emphasize that the results of the numerical solution of the BCL amplitude equation (6) are not sensitive to the noise amplitude as long as it is sufficiently small.

Figure 7

(Color online) Stability balloon for an array of $N=1230$ resonators and the same parameters as in Fig. 1, which yields $k_0\simeq 0.075$ and $\Delta Q_N\simeq 0.28$. The values of $k_n$ for $n=-1,\dots ,10$ are marked with vertical dotted lines. The thick dashed line is the neutral stability curve $g=k^2$, and the segment in which it is the lower boundary $\mid k\mid <\Delta Q_N∕2$ is marked by a solid red (dark gray) line. The black dotted and solid curves (which are almost indistinguishable) are the Eckhaus boundaries for an infinite and a finite system, respectively. The solid green (light gray) line is the saddle node $g=2k-1$ which is the lower boundary of oscillations for $k>2.47$.

Figure 8

(Color online) The three relevant Fourier amplitudes of the numerical solution of the amplitude equation (6) for $g={10}^{-4}\tau$ and the same parameters as in Fig. 7. The quasistatic values of the amplitudes (20) are plotted in thin black lines. For $\alpha ={10}^{-4}$ we expect a double phase slip from the $k_0$ pattern to the $k_2$ pattern as can be inferred from Fig. 11. The inset demonstrates the time lag at early times between the actual amplitude of the $k_0$ mode and its quasistatic value.

Figure 9

(Color online) The real part of $B(\xi ,\tau )$, obtained by a numerical integration of the BCL amplitude equation (6) for $g={10}^{-5}\tau$, using the same parameters as in Fig. 7. The initial zero state $B(\xi ,0)=0$ evolves into the $k_0$ state, which then undergoes a sequence of Eckhaus transitions as $g$ increases in time—the first two transitions involve single phase slips, while the third involves a double phase slip.

Figure 10

(Color online) The first five eigenvalues ${\lambda }_n$, plotted as a function of $g=\alpha \tau$ using the parameters of Fig. 7. As ${\lambda }_1$ turns positive the $k_0$ solution becomes Eckhaus unstable, but the selected pattern depends on the ramp rate $\alpha$ as explained in the text.

Figure 11

(Color online) The number of phase slips $Q∕\Delta Q_N$, that are observed following the Eckhaus instability of the initial $k_0$ pattern, plotted as a function of the ramp rate $\alpha$ for a linear ramp of the drive $g=\alpha \tau$. Parameters are the same as in Fig. 7. Red circles are the actual values observed in the numerical integration of the BCL amplitude equation (6). The blue line shows the predicted values from the linear analysis described in the text, where the initial amplitude of each mode, when its eigenvalue becomes positive, is taken to be ${\overline{a}}_n\left({\tau }_n\right)={10}^{-12}$, which is the accuracy of the time integration in the numerical solution of Eq. (6).