# Integrability Properties of Cubic Liénard Oscillators with Linear Damping

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Linearization via Nonlocal Transformations

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Darboux Polynomials

**Theorem**

**3.**

- 1.
- if${p}_{1}$,${p}_{2}\notin {\mathbb{Q}}^{+}\cup \left\{0\right\}$, then the polynomial$F(y,w)$is of degree at most two (with respect to y) and$$\begin{array}{c}F(y,w)={\left(\right)}_{{\left(\right)}^{w}}{\left(\right)}^{w}{s}_{2}\hfill & +\\ ,\end{array}$$
- 2.
- if${p}_{k}\in {\mathbb{Q}}^{+}$,${p}_{l}\notin {\mathbb{Q}}^{+}$, where either$k=1$,$l=2$or$k=2$,$l=1$, then the polynomial$F(y,w)$takes the form$$\begin{array}{c}F(y,w)={\left(\right)}_{{\displaystyle \prod _{j=1}^{{N}_{k}}}}{\left(\right)}^{w}{s}_{l}\hfill & +\\ ,\end{array}$$
- 3.
- if${p}_{1}\in {\mathbb{Q}}^{+}$,${p}_{2}\in {\mathbb{Q}}^{+}$, then the polynomial$F(y,w)$takes the form$$\begin{array}{c}F(y,w)={\left(\right)}_{{\displaystyle \prod _{j=1}^{{N}_{1}}}}{\displaystyle \prod _{j=1}^{{N}_{2}}}\left(\right)open="\{"\; close="\}">w-{w}_{j}^{\left(2\right)}\left(y\right)\hfill & +\\ ,\end{array}$$
- 4.
- if${p}_{1}={p}_{2}=0$, then the polynomial$F(y,w)$takes the form$$\begin{array}{c}F(y,w)=w+\frac{1}{4}{y}^{2}+4{b}_{2}y+2({b}_{1}+16{b}_{2}^{2})\end{array}$$

**Proof.**

**Remark**

**1.**

**Theorem**

**4.**

**Proof.**

**Remark**

**2.**

## 4. Conclusions and Discussion

**Theorem**

**5.**

**Proof.**

## Author Contributions

## Funding

## Conflicts of Interest

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Demina, M.; Sinelshchikov, D.
Integrability Properties of Cubic Liénard Oscillators with Linear Damping. *Symmetry* **2019**, *11*, 1378.
https://doi.org/10.3390/sym11111378

**AMA Style**

Demina M, Sinelshchikov D.
Integrability Properties of Cubic Liénard Oscillators with Linear Damping. *Symmetry*. 2019; 11(11):1378.
https://doi.org/10.3390/sym11111378

**Chicago/Turabian Style**

Demina, Maria, and Dmitry Sinelshchikov.
2019. "Integrability Properties of Cubic Liénard Oscillators with Linear Damping" *Symmetry* 11, no. 11: 1378.
https://doi.org/10.3390/sym11111378