Sensitivity of chaotic behavior to low optical frequencies of a double-beam torsional actuator

We investigate here how the optical properties at low frequencies affect the actuation dynamics and emerging chaotic behavior in a double-beam torsion actuator at nanoscale separations $\left(<200\mathrm{nm}\right)$, where the Casimir forces and torques play a major role. In fact, we take into account differences of the Casimir force due to alternative modeling of optical properties at low frequencies, where measurements are not feasible, via the Drude and plasma models, and repercussions by different material preparation conditions. For conservative autonomous actuation, bifurcation and phase portrait analysis indicate that both factors affect the stability of an actuating device in such a way that stronger Casimir forces and torques will favor increased unstable behavior. The latter will be enhanced by unbalanced application of electrostatic voltages in double-beam actuating systems. For the case of a time-periodic driving force, we use a Melnikov function and a phase plane analysis to study the emerging chaotic behavior with respect to the Drude and plasma modeling and material preparation conditions. We find indications that any factor that leads to stronger Casimir interactions will aid chaotic behavior and prevent long term prediction of the actuating dynamics. Moreover, in a double-beam actuator chaoticity will be amplified by the application of unbalanced electrostatic voltages. Therefore, the details of modeling of optical properties and the material preparations conditions must be carefully considered in the design of actuating devices at nanoscale because here Casimir forces are omnipresent and broadband type interactions.

Figure 1

Dielectric functions at imaginary frequencies $\varepsilon \left(\mathrm{i}\xi \right)$ for Au for samples 1 and 5 of [19] using the Drude and plasma models. The samples 1 and 5 from [19] have conductivity ratios ${\omega }_p^2/{\omega }_{\tau }{|}_1=1169\mathrm{eV}$ and ${\omega }_p^2/{\omega }_{\tau }{|}_5=1888.3\mathrm{eV}$, respectively. The inset shows the schematic of the double-beam torsional system.

Figure 2

Bifurcation diagrams ${\delta }_{\mathrm{Cas}}$ vs $\phi$ using the Drude and plasma models with (a) ${\delta }_v=0$; (b) ${\delta }_v=0.05$ and $p=0$ (inset: $p=0$ and ${\delta }_v=0$.5); and (c) ${\delta }_v=0.05$ and $p=1$ (inset: $p=1$ and ${\delta }_v=0.5$). The solid and dashed lines represent the stable and unstable points, respectively.

Figure 3

Bifurcation diagrams ${\delta }_v$ vs $\phi$ using the Drude and plasma models, ${\delta }_{\mathrm{Cas}}=500$, and for all studied samples: (a) $p=1$, and (b) $p=0$.

Figure 4

Contour plot of the transient times to stiction in the phase plane $d\phi /dt$ vs $\phi$ for ${\delta }_{\mathrm{Cas}}=500$ (left column) and ${\delta }_{\mathrm{Cas}}=800$ (right column), ${\delta }_v=0$, and initial conditions inside and outside of the heteroclinic orbit. The results with respect to the Drude and plasma models are shown in the left and right panels, respectively, for both samples 1 and 5.

Figure 5

Contour plot of the transient times to stiction in the phase plane $d\phi /dt$ vs $\phi$ for samples 1 and 5 with $p=1$ (left column, ${\delta }_{\mathrm{Cas}}=500$ and ${\delta }_v=0.22$), $p=0$ (right column, ${\delta }_{\mathrm{Cas}}=500$ and ${\delta }_v=0.08$), as well as initial conditions inside and outside of the heteroclinic or homoclinic orbit. The results for the Drude and plasma models are shown in the left and right panels, respectively.

Figure 6

Threshold curve $\alpha \left(=\gamma {\omega }_{0{\theta }_0}/{\tau }_0\right)$ vs. driving frequency $\omega /{\omega }_{\mathrm{o}}$ (with ${\omega }_{\mathrm{o}}$ the natural frequency of the system). The area bellow the curve corresponds to parameters that could lead to chaotic motion with ${\delta }_{\mathrm{Cas}}=500$ and ${\delta }_v=0$ for both the Drude and plasma models: (a) sample 1, and (b) sample 5.

Figure 7

Threshold curve $\alpha \left(=\gamma {\omega }_{0{\theta }_0}/{\tau }_0\right)$ vs driving frequency $\omega /{\omega }_{\mathrm{o}}$ (with ${\omega }_{\mathrm{o}}$ the natural frequency of the system). The area below the curve corresponds to parameters that could lead to chaotic motion with ${\delta }_{\mathrm{Cas}}=500, {\delta }_v=0.08$, and $p=0$ (unbalanced case) for both the Drude and plasma models: (a) sample 1, and (b) sample 5.

Figure 8

Threshold curve $\alpha \left(=\gamma {\omega }_{0{\theta }_0}/{\tau }_0\right)$ vs driving frequency $\omega /{\omega }_{\mathrm{o}}$ (with ${\omega }_{\mathrm{o}}$ the natural frequency of the system). The area below the curve corresponds to parameters that can lead to chaotic motion with ${\delta }_{\mathrm{Cas}}=500, {\delta }_v=0.22$, and $p=1$ (balanced case) for both the Drude and plasma models: (a) sample 1, (b) sample 5.

Figure 9

Contour plot of the transient times to stiction in the phase plane $d\phi /dt$ vs $\phi$ for the nonconservative system with $\alpha =0.5$ and $\omega /{\omega }_{\mathrm{o}}=0.2$: $p=1$ (balanced case, left column for ${\delta }_{\mathrm{Cas}}=500$ and ${\delta }_v=0.22$), and $p=0$ (unbalanced case, right column for ${\delta }_{\mathrm{Cas}}=500$ and ${\delta }_v=0.08$) for samples 1 and 5 with plasma model (right column) and Drude model (left column). The red (light gray) elliptical shape central area corresponds to stable actuation.

Figure 10

Contour plot of the transient times to stiction in the phase plane $d\phi /dt$ vs $\phi$ for the nonconservative system with $\alpha =0.5, \omega /{\omega }_{\mathrm{o}}=0.2, {\delta }_{\mathrm{Cas}}=500, {\delta }_v=0.08$, and considering the plasma model for sample 1 and 5: $p=0$ (unbalanced case, right column), and $p=1$ (balanced case, left column).

Figure 11

Contour plot of the transient times to stiction in the phase plane $d\phi /dt$ vs $\phi$ for the nonconservative system with $\alpha =0.5, \omega /{\omega }_{\mathrm{o}}=0.2, {\delta }_{\mathrm{Cas}}=500, {\delta }_v=0.08$ for sample 5 with respect to Drude and plasma models: $p=0$ in right panel (unbalanced case), and $p=1$ in left panel (balanced case).

Figure 12

Contour plot of the transient times to stiction in the phase plane $d\phi /dt$ vs $\phi$ for the nonconservative system with $\alpha =0.5, \omega /{\omega }_{\mathrm{o}}=0.2, {\delta }_v=0.22$, sample 5, and $p=1$ (balanced situation): ${\delta }_{\mathrm{Cas}}=500$ (right panel), and ${\delta }_{\mathrm{Cas}}=430$ (left panel). The red (light gray) elliptical shape central area corresponds to stable actuation.

Figure 13

Contour plot of the transient times to stiction in the phase plane $d\phi /dt$ vs $\phi$ for sample 5 of the nonconservative system with $\alpha =0.02, \omega /{\omega }_{\mathrm{o}}=0.5, p=0$ (unbalanced case), ${\delta }_{\mathrm{Cas}}=430$, and ${\delta }_v=0.09$ for different oscillations cycles as indicated. Plasma model: right column, and Drude model: left column.

Figure 14

Contour plot of the transient times to stiction in the phase plane $d\phi /dt$ vs $\phi$ for sample 5 of the nonconservative system with $\alpha =0.02, \omega /{\omega }_{\mathrm{o}}=0.35, {\delta }_{\mathrm{Cas}}=430$, and ${\delta }_v=0.22$ for different oscillation cycles. Plasma model: right column, and Drude model left column.

Figure 15

Area of stable motion $S$ (large diameter $\times$ short diameter) vs $N$ (number of oscillation) for sample 5 for both the Drude and plasma models. (a) $p=0$ (electrostatically unbalanced case), and (b) $p=1$ (electrostatically balanced case).