# On Some Reversible Cubic Systems

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Preliminaries

**Theorem**

**1.**

## 3. Some Center Conditions

**Theorem**

**2.**

**Proof.**

`Eliminate`of Mathematica from the remaining equations ${a}_{0},{b}_{0},{b}_{2},k,m,p,q$ we obtain the ideal

`radical`of the compute algebra system Singular [34] the radical of U we obtain the ideal ${I}_{1}$ given in the statement of the theorem.

`minAssGTZ`[35] of Singular we computed the decomposition of the obtained ideal and found two components (output provided by Singular is presented at http://www.camtp.uni-mb.si/camtp/barbara/idealVed (accessed on 6 March 2021)). Performing the rational reconstruction of the first component with the algorithm of [35] we obtained the ideal ${\widehat{I}}_{3}$ given in Appendix B.

`eliminate`of Singular we computed the ideal

`minAssGTZ`we obtain the ideals ${\widehat{I}}_{4}$ and ${\widehat{I}}_{5}$ given in Appendix D.

`eliminate`of Singular we eliminate from the remaining polynomial the variables m and n obtaining an ideal whose variety is defined by (17) and (18). This means that systems from family (10) correspond to systems from $\mathbf{V}\left({I}_{5}\right)$ for which conditions (17) and (18) are fulfilled.

`genus`of the library

`normal.lib`[34] of the computer algebra system Singular we found that for

`paraPlaneCurve`of the library

`paraplanecurves.lib`[37] we obtained that the curve on the variety ${V}_{6}$ is defined parametrically in the following way

**Remark**

**1.**

`eliminate`of Singular we eliminated t from ${J}_{2}$ and then used the radical membership test [14,33].

**Remark**

**2.**

## 4. Orbital Reversibility in Subfamilies of (6)

**Theorem**

**3.**

- (1)
- If $F\left(p\right)=-1$ and $det\left(D\mathcal{X}\right(p\left)\right)>0,$ then p is a center of $\mathcal{X}$.
- (2)
- If $F\left(p\right)=-1$ and $det\left(D\mathcal{X}\right(p\left)\right)<0$, then p is a saddle of $\mathcal{X}$.
- (3)
- If $F\left(p\right)=1$, then $Fix\phantom{\rule{0.166667em}{0ex}}\phi \cap U$ is invariant under the flow of $\mathcal{X}$.

**Lemma**

**1.**

**Proposition**

**1.**

**Proof.**

**Remark**

**4.**

## 5. Bifurcations of Limit Cycles

**Theorem**

**4.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**5.**

${V}_{1}$ | ${V}_{2}$ | ${V}_{3}$ | ${V}_{4}$ | ${V}_{5}$ | |

$\mathsf{A}$ | 3 | 2 | 4 | 2 | 2 |

$\mathsf{B}$ | 4 | 3 | 5 | 3 | 3 |

$\mathsf{C}$ | 6 | 3 | 6 | 3 | 5 |

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## Appendix D

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Arcet, B.; Romanovski, V.G.
On Some Reversible Cubic Systems. *Mathematics* **2021**, *9*, 1446.
https://doi.org/10.3390/math9121446

**AMA Style**

Arcet B, Romanovski VG.
On Some Reversible Cubic Systems. *Mathematics*. 2021; 9(12):1446.
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**Chicago/Turabian Style**

Arcet, Barbara, and Valery G. Romanovski.
2021. "On Some Reversible Cubic Systems" *Mathematics* 9, no. 12: 1446.
https://doi.org/10.3390/math9121446