Nonlinear dynamics and chaos in two coupled nanomechanical resonators

Two elastically coupled nanomechanical resonators driven independently near their resonance frequencies show intricate nonlinear dynamics. The dynamics provide a scheme for realizing a nanomechanical system with tunable frequency and nonlinear properties. For large vibration amplitudes, the system develops spontaneous oscillations of amplitude modulation that also show period-doubling transitions and chaos. The complex nonlinear dynamics are quantitatively predicted by a simple theoretical model.

Figure 1

(Color online) (a) SEM image of the system (beam dimensions: $6\mu \text{m}\times 500\text{nm}\times 200\text{nm}$). (b) Up (dark, blue) and down (light, yellow) frequency sweeps of the amplitude $\left|A\right|$ of the response for a single drive of strength 4.3 times critical as a function of the frequency relative to the linear resonance frequency ${\omega }_0$ and scaled by the width $\gamma$. The amplitude is plotted in units of the maximum amplitude at the critical drive strength. (c) Frequency shift of the weakly driven first mode as a function of the displacement amplitude $\left|A_{\text{II}}\right|$ of the strongly driven second mode: points and solid line—experiment; dashed line—small amplitude theory.

Figure 2

(Color online) Frequency sweeps of the first mode response for increasing amplitudes of the second mode: the dark (blue) lines are for upward sweeps; the light (yellow) line in (a) is a downward sweep, showing the hysteresis and amplitude jumps in this case. There are no hysteretic jumps for (b) and (c), and so the downward sweeps are not shown.

Figure 3

(Color online) First mode frequency response as in Fig. 2 for increasing values of the second mode drive frequency but for stronger driving of the first mode (both modes are driven at about four times the critical strength). The top plots are experimental measurements while the bottom ones are theoretical simulations. The second mode response was monitored but is not shown.

Figure 4

Complex dynamics of two strongly coupled nanomechanical resonators. The three rows correspond to different input parameters (drive frequencies and amplitudes). The first column shows the theoretical calculation of the phase portrait, the second shows its experimental measurement, and the third column shows the corresponding experimental power spectrum of the optical measurement near one of the drive frequencies.