# Nonlinear Dynamics of a New Class of Micro-Electromechanical Oscillators—Open Problems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Results in the Light of the Melnikov’s Approach

#### 3.1. The Case $n=m=3$ and $N=1$

**Proposition 1.**

**Remark 1.**

#### 3.2. The Case n = m = 3 and N = 2

**Proposition 2.**

#### 3.3. The Case n = m = 3 and N = 3

**Proposition 3.**

## 4. Open Problems

**Problem 1.**

**Problem 2.**

## 5. Some Simulations

**Problem 3.**

#### 5.1. Parameters Generated by a Probability Distribution

## 6. An Example with a Relatively Large Number of Terms: n = 5, m = 5

## 7. Some Difficulties That the User Encounters When Using Computer Algebraic Systems for Scientific Calculation

**Proposition 4.**

## 8. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Double homoclinic orbit [27].

**Figure 3.**The nonlinear Equation (6): (

**a**) for $\alpha =1.39627$, ${a}_{0}=1.2$, $a=1$, $\omega =2.2$, ${g}_{1}=0.7$, $M\left({t}_{0}\right)$ has root ${t}_{0}\approx 0.0003209$ with multiplicity two; (

**b**) for $\alpha =1.39627$, ${a}_{0}=1.15$, $a=1$, $\omega =2.5$, ${q}_{1}=0.6$, $M\left({t}_{0}\right)$ has no roots.

**Figure 4.**The nonlinear Equation (8): (

**a**) for $\alpha =1.7$, ${a}_{0}=0.8$, $a=1$, $\omega =0.75$, ${g}_{1}=0.7$, ${g}_{2}=0.3$, ${g}_{3}=0.1$, $M\left({t}_{0}\right)$ has root ${t}_{0}\approx 0.144$; (

**b**) for $\alpha =1.95$, ${a}_{0}=0.7$, $a=1$, $\omega =0.6$, ${g}_{1}=0.8$, ${g}_{2}=0.4$, ${g}_{3}=0.15$, $M\left({t}_{0}\right)$ has no roots.

**Figure 10.**Potential energy $V\left(x\right)$ for $b=-1,\phantom{\rule{0.222222em}{0ex}}{b}_{1}=-0.7,\phantom{\rule{0.222222em}{0ex}}{b}_{0}=0.1$.

**Figure 11.**For $b=-0.4,\phantom{\rule{0.222222em}{0ex}}{b}_{1}=-0.7,\phantom{\rule{0.222222em}{0ex}}{b}_{0}=0.1$ (

**a**) homoclinic orbits; (

**b**) heteroclinic orbits.

**Figure 14.**The equation $M\left({t}_{0}\right)=0$ (Proposition 5.) for fixed parameters: (

**a**) $p=2$, $N=3$, $A=1.01$, $\omega =0.35$, ${g}_{1}=0.5$, ${g}_{2}=0.95$, ${g}_{3}=0.8$; (

**b**) $A=0.1$, $\omega =0.9$, ${g}_{1}=0.01$, ${g}_{2}=0.02$, ${g}_{3}=0.015$, $M\left({t}_{0}\right)$ has no roots.

**Figure 15.**Melnikov function $M\left(t\right)$ (thick) for $\alpha =1.95$, ${a}_{0}=0.76$, $a=1$, $\omega =0.6$, ${g}_{1}=0.8$, ${g}_{2}=0.4$, ${g}_{3}=0.15$ in interval [0,10] (see Proposition 3).

**Figure 16.**Melnikov function $M\left(t\right)$ for $A=0.011$, $\omega =0.9$, ${g}_{1}=-0.01$, ${g}_{2}=-0.02$, ${g}_{3}=-0.03$ in interval $[-5,5]$ (see Proposition 4).

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**MDPI and ACS Style**

Kyurkchiev, N.; Zaevski, T.; Iliev, A.; Kyurkchiev, V.; Rahnev, A.
Nonlinear Dynamics of a New Class of Micro-Electromechanical Oscillators—Open Problems. *Symmetry* **2024**, *16*, 253.
https://doi.org/10.3390/sym16020253

**AMA Style**

Kyurkchiev N, Zaevski T, Iliev A, Kyurkchiev V, Rahnev A.
Nonlinear Dynamics of a New Class of Micro-Electromechanical Oscillators—Open Problems. *Symmetry*. 2024; 16(2):253.
https://doi.org/10.3390/sym16020253

**Chicago/Turabian Style**

Kyurkchiev, Nikolay, Tsvetelin Zaevski, Anton Iliev, Vesselin Kyurkchiev, and Asen Rahnev.
2024. "Nonlinear Dynamics of a New Class of Micro-Electromechanical Oscillators—Open Problems" *Symmetry* 16, no. 2: 253.
https://doi.org/10.3390/sym16020253