Generic Rotating-Frame-Based Approach to Chaos Generation in Nonlinear Micro- and Nanoelectromechanical System Resonators

This Letter provides a low-power method for chaos generation that is generally applicable to nonlinear micro- and nanoelectromechanical systems (MNEMS) resonators. The approach taken is independent of the material, scale, design, and actuation of the device in question; it simply assumes a good quality factor and a Duffing type nonlinearity, features that are commonplace to MNEMS resonators. The approach models the rotating-frame dynamics to analytically constrain the parameter space required for chaos generation. By leveraging these common properties of MNEMS devices, a period-doubling route to chaos is generated using smaller forcing than typically reported in the literature.

Figure 1

(a) Bistability map plotted as a function of dimensionless force and detuning, showing the region of bistability for a lossless driven Duffing resonator (gray area), and for a low-loss ($Q=1000$) Duffing resonator (area between the dashed blue lines). (b) Amplitude (arbitrary units) versus detuning response of a lossless Duffing taken for $F_1\sqrt{\alpha }={10}^{-4}$. The corresponding phase-space plot (also in arbitrary units) for a detuning of $\delta =2.5\times {10}^{-3}$ is shown in (c). The stable fixed points and the saddle point are shown as black and green dots, respectively, and the black traces correspond to the homoclinic orbits. Small amplitude libration orbits around the high-amplitude branch (blue) and low-amplitude branch (red) are shown. $\alpha$ is the Duffing parameter.

Figure 2

(a) Experimental lock-in amplifier data for the frequency response obtained by a single tone sweep showing the linear (black trace $100\mathrm{m}{\mathrm{V}}_{\mathrm{PP}}$) and the Duffing regimes ($3{\mathrm{V}}_{\mathrm{PP}}$); the latter shows bistability upon performing a forward (blue trace) and a backward sweep (red trace). $H$ denotes the relative amplitude response, expressed in mV per V drive. (b) Scatter plot of periodically sampled libration oscillation under the effect of a two-tone excitation, with one fixed tone (indicated by $F_1$ having ${\omega }_1/2\pi =1566.5\mathrm{kHz}$ and $F_1=3{\mathrm{V}}_{\mathrm{PP}}$) and one swept tone ($F_2=2.1{\mathrm{V}}_{\mathrm{PP}}$, $1558\mathrm{kHz}<{\omega }_2/2\pi <1576\mathrm{kHz}$), shown for the lower (red) and higher (blue) amplitude branches. Period 1, period 2, and chaotic oscillations are detected and shown in (c)–(e), respectively, for both the high (blue) and low (red) amplitude branches. The black dots correspond to the experimentally-obtained fixed points, and the crosses correspond to the calculated saddle point.

Figure 3

Numerical simulations obtained for $\delta =4.2\times {10}^{-3}$ (i.e., ${\omega }_1/2\pi =1566.5\mathrm{kHz}$), and $F_1\sqrt{\alpha }=1.5\times {10}^{-4}$ (i.e., ${\mathrm{F}}_1=3{\mathrm{V}}_{\mathrm{PP}}$). (a) Basins of attraction under the effect of $F_2=0.6F_1$, $\mathrm{\Omega }=3.275\times {10}^{-3}$ for the high-branch, and $F_2=0.5F_1$, $\mathrm{\Omega }=-2.7\times {10}^{-3}$ for the low-branch showing that only one of the two basins is disturbed depending on the value of $\mathrm{\Omega }$. The white dots and crosses mark the location of the stable and saddle points in the unperturbed system, respectively. (b) Two-dimensional maps (both panels) showing the location of autocorrelation peak as a function of detuning and forcing ($\mathrm{\Omega }$, $F_2$). Values greater than 1 (red and bright areas) indicate period-doubling bifurcations and chaos. The experimentally obtained bifurcation areas are equally shown (delineated by the dashed lines). The area bounded by the analytical model, i.e., $F_{2\mathrm{C}}$, is shown as the solid yellow lines. The solid vertical lines indicate where in the ($F_2$, $\mathrm{\Omega }$) parameter space the basins in (a) are located.

Figure 4

Comparison between numerical (dots) and analytical (solid lines) values of $F_{2\min }$ (b) and ${\mathrm{\Omega }}_{\min }$ (c) required for $P2$ and chaos for both the low amplitude branch (red) and high amplitude branch (blue) of the Duffing resonator shown in (a). For both numerical and analytical data, $P2$ only appears in the range where bistability exists (indicated by the dashed vertical gray lines, and denoted $A$ and $B$, respectively). As the high amplitude branch approaches the saddle-node bifurcation its libration motion becomes more unstable, hence the data points do not reach the saddle-node bifurcation.