# Global Bifurcation Behaviors and Control in a Class of Bilateral MEMS Resonators

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Unperturbed Dynamics

_{AC}<< V

_{b}, the parameters μ and β in Equation (5) are both small, the concerned terms can be considered as the perturbed ones. Thus, the unperturbed system of Equation (5) is

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

## 3. Global Bifurcations and Complex Dynamics

#### 3.1. Homoclinic Bifurcation Behavior

#### 3.2. Heteroclinic Bifurcation Behavior

## 4. Control of Complex Dynamics via Delayed Feedback

#### 4.1. Delayed Position Feedback

#### 4.1.1. Control of Chaos

#### 4.1.2. Control of Pull-In Instability

#### 4.2. Delayed Velocity Feedback

#### 4.2.1. Suppression of Chaos

#### 4.2.2. Suppression of Pull-In Instability

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Comparison of homoclinic orbits of the system (6) where the solid curve represents the exact model based on Hamiltonian and the sign + shows the orbits in explicit functions of $\phi $.

**Figure 5.**Dynamic behavior of the system (5) for ${V}_{AC}=0.16\mathrm{V},{V}_{\mathrm{b}}=3.8\mathrm{V},\omega =0.5$: (

**a**) phase portrait; (

**b**) Poincáre map.

**Figure 6.**Comparison of heteroclinic orbits of the system (6) where the solid curve represents the exact model based on Hamiltonian and the sign + shows the orbits in explicit functions of $\psi $.

**Figure 7.**Safe basins of the system (5) under different values of AC voltage amplitude: (

**a**) ${V}_{AC}=0\mathrm{V}$; (

**b**) ${V}_{AC}=0.15\mathrm{V}$; (

**c**) ${V}_{AC}=0.30\mathrm{V}$; (

**d**) ${V}_{AC}=0.50\mathrm{V}$; (

**e**) ${V}_{AC}=0.70\mathrm{V}$; (

**f**) ${V}_{AC}=1.05\mathrm{V}$.

**Figure 9.**Variation of ${V}_{AC}$ threshold for homoclinic bifurcation with $\tau $ when ${g}_{p}=0.2$.

**Figure 10.**Dynamic behavior of the delayed system (39) when ${g}_{p}=0.2$ and ${V}_{AC}=0.16\mathrm{V}$ (

**a**) Bifurcation with τ in Poincáre map (

**b**) Phase map when τ = 0.12.

**Figure 12.**Safe basins of the controlled system (39) under different values of ${V}_{AC}$ and τ: (

**a**) ${V}_{AC}=0.46\mathrm{V},\hspace{1em}\tau =0$; (

**b**) ${V}_{AC}=0.46\mathrm{V},\hspace{1em}\tau =0.55$; (

**c**) ${V}_{AC}=0.46\mathrm{V},\hspace{1em}\tau =0.75$; (

**d**) ${V}_{AC}=0.52\mathrm{V},\hspace{1em}\tau =0$; (

**e**) ${V}_{AC}=0.52\mathrm{V},\hspace{1em}\tau =0.55$; (

**f**) ${V}_{AC}=0.52\mathrm{V},\hspace{1em}\tau =0.75$; (

**g**) ${V}_{AC}=0.81\mathrm{V},\hspace{1em}\tau =0$; (

**h**) ${V}_{AC}=0.81\mathrm{V},\hspace{1em}\tau =0.55$; (

**i**) ${V}_{AC}=0.81\mathrm{V},\hspace{1em}\tau =0.75$.

**Figure 13.**Bifurcation of the delayed-velocity-feedback controlled system (40) with $\tau $ in Poincaré map for ${g}_{v}=0.2$ and ${V}_{AC}=0.16\mathrm{V}$.

**Figure 14.**Variation of AC voltage threshold for heteroclinic bifurcation with $\tau $ when g

_{v}= 0.2.

**Figure 15.**Safe basins of velocity-feedback-controlled system (40) under different ${V}_{AC}$ and τ: (

**a**) ${V}_{AC}=0.46\mathrm{V},\hspace{1em}\tau =0$; (

**b**) ${V}_{AC}=0.46\mathrm{V},\hspace{1em}\tau =1.6$; (

**c**) ${V}_{AC}=0.46\mathrm{V},\hspace{1em}\tau =2$; (

**d**) ${V}_{AC}=0.52\mathrm{V},\hspace{1em}\tau =0$; (

**e**) ${V}_{AC}=0.52\mathrm{V},\hspace{1em}\tau =1.6$; (

**f**) ${V}_{AC}=0.52\mathrm{V},\hspace{1em}\tau =2$; (

**g**) ${V}_{AC}=0.81\mathrm{V},\hspace{1em}\tau =0$; (

**h**) ${V}_{AC}=0.81\mathrm{V},\hspace{1em}\tau =1.6$; (

**i**) ${V}_{AC}=0.81\mathrm{V},\hspace{1em}\tau =2.0$.

Parameter | Symbol |
---|---|

Equivalent mass of the proof mass (kg) | m |

Viscous damping coefficient in the high vacuum environment (N·s/m) | c |

Linear stiffness coefficient (N/m) | ${k}_{1}$ |

Cubic stiffness term (N/m^{3}) | ${k}_{3}$ |

Capacitance of each parallel plate at rest ($\mathrm{Fm}$) | ${A}_{0}$ |

Initial gap width between the two neighboring parallel plates (m) | d |

DC bias voltage (V) | ${V}_{b}$ |

Frequency of AC voltage (HZ) | Ω |

Amplitude of AC voltage (V) | ${V}_{AC}$ |

Time | t |

Displacement of the proof mass at time t | $z$ |

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**MDPI and ACS Style**

Zhu, Y.; Shang, H.
Global Bifurcation Behaviors and Control in a Class of Bilateral MEMS Resonators. *Fractal Fract.* **2022**, *6*, 538.
https://doi.org/10.3390/fractalfract6100538

**AMA Style**

Zhu Y, Shang H.
Global Bifurcation Behaviors and Control in a Class of Bilateral MEMS Resonators. *Fractal and Fractional*. 2022; 6(10):538.
https://doi.org/10.3390/fractalfract6100538

**Chicago/Turabian Style**

Zhu, Yijun, and Huilin Shang.
2022. "Global Bifurcation Behaviors and Control in a Class of Bilateral MEMS Resonators" *Fractal and Fractional* 6, no. 10: 538.
https://doi.org/10.3390/fractalfract6100538