# Chenciner Bifurcation Presenting a Further Degree of Degeneration

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## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Results

#### 3.1. Degree of the Second Bifurcation Curve Is One in the Truncated Version

**Remark**

**1.**

- For $p>0,\phantom{\rule{0.277778em}{0ex}}q\ne 0$, there is one real root ${e}_{1},$ and two complex conjugated ones;
- For $p<0,\phantom{\rule{0.277778em}{0ex}}q=0$, there is a triple root ${e}_{1}$;
- For $p<0,\phantom{\rule{0.277778em}{0ex}}r>0$, there is one real root ${e}_{1},$ and two complex conjugated ones;
- For $p<0,\phantom{\rule{0.277778em}{0ex}}r=0$, there are three real roots, one simple ${e}_{1}$, and two common;
- For $p<0,\phantom{\rule{0.277778em}{0ex}}r<0$, there are three real different roots ${e}_{1}<{e}_{2}<{e}_{3}.$

**Lemma**

**1.**

- ${L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}k>0$
- ${L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}k<0$
- ${L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}k>0$
- ${L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}k<0.$

**Remark**

**2.**

**Remark**

**3.**

- if $a>0,\phantom{\rule{0.277778em}{0ex}}k>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a>0,\phantom{\rule{0.277778em}{0ex}}k<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}k>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}k<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$, we get Figure 3a;
- if $a>0,\phantom{\rule{0.277778em}{0ex}}k>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a>0,\phantom{\rule{0.277778em}{0ex}}k<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}k>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}k<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$, we get Figure 3b.

#### 3.2. Degree of the Second Bifurcation Curve Is Two

- I
- ${e}_{1}<{m}_{1}<{m}_{2}<{e}_{2}<{e}_{3}$;
- II
- ${e}_{1}<{m}_{1}<{e}_{2}<{m}_{2}<{e}_{3}$;
- III
- ${e}_{1}<{m}_{1}<{e}_{2}<{e}_{3}<{m}_{2}.$

**Remark**

**4.**

- (1)
- If $a>0,\phantom{\rule{0.277778em}{0ex}}h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$, then we get the Figure 8a.
- (2)
- If $a>0,\phantom{\rule{0.277778em}{0ex}}h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$, then we get the Figure 8b.
- (3)
- If $a>0,\phantom{\rule{0.277778em}{0ex}}h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$, then we get the Figure 9a.
- (4)
- If $a>0,\phantom{\rule{0.277778em}{0ex}}h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$, then we get the Figure 9b.

- ${\Delta}_{2}<0$ implies $sign\phantom{\rule{0.277778em}{0ex}}{\beta}_{2}\left(\alpha \right)=sign\left(h\right);$
- ${\Delta}_{2}>0$, then there is ${m}_{1},\phantom{\rule{0.277778em}{0ex}}{m}_{2}$, two distinct real roots of ${\beta}_{2}\left(\alpha \right)=0$ and$$sign\phantom{\rule{0.277778em}{0ex}}{\beta}_{2}\left(\alpha \right)=\left\{\begin{array}{cc}sign\left(h\right),\hfill & \mathrm{if}\phantom{\rule{1.em}{0ex}}m\in (-\infty ,{m}_{1})\cup ({m}_{2},\infty )\hfill \\ -sign\left(h\right),\hfill & \mathrm{if}\phantom{\rule{1.em}{0ex}}m\in ({m}_{1},{m}_{2}).\hfill \end{array}\right.$$

**Remark**

**5.**

- If $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h>0$, see Figure 10a;
- If $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$, see Figure 10b;
- If $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$, see Figure 11a;
- If $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$, see Figure 11b.

**Remark**

**6.**

- If ${e}_{1}<{m}_{1}<{m}_{2}$ and $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h>0$ or if ${m}_{1}<{e}_{1}<{m}_{2}$ and $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$, then will obtain Figure 10a.
- If ${e}_{1}<{m}_{1}<{m}_{2}$ and $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$ or if ${m}_{1}<{e}_{1}<{m}_{2}$ and $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$, then will obtain Figure 10b.
- If ${e}_{1}<{m}_{1}<{m}_{2}$ and $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$ or if ${m}_{1}<{e}_{1}<{m}_{2}$ and $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h>0$, then will obtain Figure 11a.
- If ${e}_{1}<{m}_{1}<{m}_{2}$ and $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$ or if ${m}_{1}<{e}_{1}<{m}_{2}$ and $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$, then will obtain Figure 11b.

#### 3.3. Numerical Simulations

## 4. Discussions and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Chenciner Bifurcations

## Appendix B. Literature Review

**Theorem**

**A1.**

**Theorem**

**A2.**

- (a)
- one invariant unstable circle ${\rho}_{n}=\sqrt{{y}_{1}}$ if ${L}_{0}>0$ and ${\beta}_{1}\left(\alpha \right)<0;$
- (b)
- one invariant stable circle ${\rho}_{n}=\sqrt{{y}_{2}}$ if ${L}_{0}<0$ and ${\beta}_{1}\left(\alpha \right)>0;$
- (c)
- two invariant circles, ${\rho}_{n}=\sqrt{{y}_{1}}$ unstable and ${\rho}_{n}=\sqrt{{y}_{2}}$ stable, if ${L}_{0}>0,$${\beta}_{1}\left(\alpha \right)>0,$${\beta}_{2}\left(\alpha \right)<0$ or ${L}_{0}<0,$${\beta}_{1}\left(\alpha \right)<0,$${\beta}_{2}\left(\alpha \right)>0;$ in addition, ${y}_{1}<{y}_{2}$ if ${L}_{0}<0$ and ${y}_{2}<{y}_{1}$ if ${L}_{0}>0;$
- (d)
- no invariant circles if ${L}_{0}>0,$${\beta}_{1}\left(\alpha \right)>0,$${\beta}_{2}\left(\alpha \right)>0$ or ${L}_{0}<0,$${\beta}_{1}\left(\alpha \right)<0,$${\beta}_{2}\left(\alpha \right)<0.$

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**Figure 1.**Bifurcation diagrams when $p<0,\phantom{\rule{0.277778em}{0ex}}r<0$, and $hk>0$: (

**a**) ${L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}k>0$; (

**b**) ${L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}k<0$.

**Figure 2.**Bifurcation diagrams when $p<0,\phantom{\rule{0.277778em}{0ex}}r<0$, and $hk>0$: (

**a**) ${L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}k>0$; (

**b**) ${L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}k<0$.

**Figure 3.**Bifurcation diagrams when $p>0$ or ($p<0$ and $r>0$) and $hk>0$: (

**a**) $a>0,\phantom{\rule{0.277778em}{0ex}}k>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a>0,\phantom{\rule{0.277778em}{0ex}}k<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}k>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}k<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$; (

**b**) $a>0,\phantom{\rule{0.277778em}{0ex}}k>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a>0,\phantom{\rule{0.277778em}{0ex}}k<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}k>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}k<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$.

**Figure 4.**Bifurcation diagrams in the Case I when $p<0,\phantom{\rule{0.277778em}{0ex}}r<0$, and ${\Delta}_{2}>0$: (

**a**) ${L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h>0$; (

**b**) ${L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$.

**Figure 5.**Bifurcation diagrams in the Case I when $p<0,\phantom{\rule{0.277778em}{0ex}}r<0$, and ${\Delta}_{2}>0$: (

**a**) ${L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$; (

**b**) ${L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$.

**Figure 6.**Bifurcation diagrams in the Case II and III when $p<0,\phantom{\rule{0.277778em}{0ex}}r<0$, and ${\Delta}_{2}>0$: (

**a**) ${L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h>0$ or $h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$; (

**b**) ${L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$ or ${L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$.

**Figure 7.**Bifurcation diagrams in the Case II and III when $p<0,\phantom{\rule{0.277778em}{0ex}}r<0$, and ${\Delta}_{2}>0$: (

**a**) ${L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$; (

**b**) $h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$.

**Figure 8.**Bifurcation diagrams when $p<0,\phantom{\rule{0.277778em}{0ex}}r<0$, and ${\Delta}_{2}<0$: (

**a**) $a>0,\phantom{\rule{0.277778em}{0ex}}h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$; (

**b**) $a>0,\phantom{\rule{0.277778em}{0ex}}h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}h>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$.

**Figure 9.**Bifurcation diagrams when $p<0,\phantom{\rule{0.277778em}{0ex}}r<0$, and ${\Delta}_{2}<0$: (

**a**) $a>0,\phantom{\rule{0.277778em}{0ex}}h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0$; (

**b**) $a>0,\phantom{\rule{0.277778em}{0ex}}h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}h<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0$.

**Figure 10.**Bifurcation diagrams when $p>0$ or ($p<0$ and $r>0$): (

**a**) $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h>0$; (

**b**) $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h>0$.

**Figure 11.**Bifurcation diagrams when $p>0$ or ($p<0$ and $r>0$): (

**a**) $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}>0,\phantom{\rule{0.277778em}{0ex}}h<0$; (

**b**) $a>0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$ or $a<0,\phantom{\rule{0.277778em}{0ex}}{L}_{0}<0,\phantom{\rule{0.277778em}{0ex}}h<0$.

**Figure 12.**Numerical simulation for the map (9) when ${\beta}_{1}\left(\alpha \right)=2{\alpha}_{1}^{3}+{\alpha}_{2}^{3}+{\alpha}_{1}^{2}{\alpha}_{2}$, ${\beta}_{2}\left(\alpha \right)=-{\alpha}_{1}^{2}-{\alpha}_{1}{\alpha}_{2}-{\alpha}_{2}^{2}$, with ${\alpha}_{1}=0.1,\phantom{\rule{0.277778em}{0ex}}{\alpha}_{2}=0.1$: (

**a**) blue orbit starts from $({\rho}_{1},{\phi}_{1})=(0.01,0)$; (

**b**) red orbit starts from $({\rho}_{1},{\phi}_{1})=(0.03,0)$.

**Figure 13.**Numerical simulation for the map (9) when ${\beta}_{1}\left(\alpha \right)=2{\alpha}_{1}^{3}+{\alpha}_{2}^{3}+{\alpha}_{1}^{2}{\alpha}_{2}$, ${\beta}_{2}\left(\alpha \right)=-{\alpha}_{1}^{2}-{\alpha}_{1}{\alpha}_{2}-{\alpha}_{2}^{2}$: (

**a**) the three orbits are represented here with $({\rho}_{1},{\phi}_{1})=(0.01,0)$, $({\rho}_{1},{\phi}_{1})=(0.03,0)$ and $({\rho}_{1},{\phi}_{1})=(0.3,0)$, respectively, and ${\alpha}_{1}=0.1,\phantom{\rule{0.277778em}{0ex}}{\alpha}_{2}=0.1$; (

**b**) the three orbits are represented here with $({\rho}_{1},{\phi}_{1})=(0.183,0)$, $({\rho}_{1},{\phi}_{1})=(0.16,0)$ and $({\rho}_{1},{\phi}_{1})=(0.14,0)$, respectively, and ${\alpha}_{1}=0.5$, ${\alpha}_{2}=-0.513$.

**Figure 14.**The discrete sequence ${\rho}_{n}$ given by the map (9) in the plane $\left(nO{\rho}_{n}\right)$ when ${\beta}_{1}\left(\alpha \right)=2{\alpha}_{1}^{3}+{\alpha}_{2}^{3}+{\alpha}_{1}^{2}{\alpha}_{2}$, ${\beta}_{2}\left(\alpha \right)=-{\alpha}_{1}^{2}-{\alpha}_{1}{\alpha}_{2}-{\alpha}_{2}^{2}$: (

**a**) when $({\rho}_{1},{\phi}_{1})=(0.01,0)$, $({\rho}_{1},{\phi}_{1})=(0.03,0)$, $({\rho}_{1},{\phi}_{1})=(0.284,0)$ and ${a}_{1}=0.1,\phantom{\rule{0.277778em}{0ex}}{a}_{2}=0.1$; (

**b**) when starting points are as in Figure 13b.

**Figure 15.**Numerical simulations for the map (9) when ${\beta}_{1}\left(\alpha \right)={\alpha}_{1}^{3}+2{\alpha}_{2}^{3}-{\alpha}_{1}{\alpha}_{2}^{2}-{\alpha}_{1}^{2}{\alpha}_{2}$, ${\beta}_{2}\left(\alpha \right)=-{\alpha}_{1}-3{\alpha}_{2}$ and $({\alpha}_{1},{\alpha}_{2})=(0.1,-0.1)$: (

**a**) four orbits corresponding to $({\rho}_{1},{\phi}_{1})=(0.06,0)$ (the orbit in red), $({\rho}_{1},{\phi}_{1})=(0.187,0)$ (the orbit in blue), $({\rho}_{1},{\phi}_{1})=(0.0716,0)$ (the orbit in magenta), $({\rho}_{1},{\phi}_{1})=\left(0.06\right)$ (the orbit in green); (

**b**) the sequence ${\rho}_{n}$ in the plane $\left(nO{\rho}_{n}\right)$ corresponding to the green orbit, when $N=6000$ from (

**a**).

**Figure 16.**Numerical simulations for the map (9): (

**a**) sequence ${x}_{n}$ in the plane $\left(nO\rho \right)$ from Figure 15b; (

**b**) numerical simulations for the map (9) when ${\beta}_{1}\left(\alpha \right)=2{\alpha}_{1}^{3}+{\alpha}_{2}^{3}+{\alpha}_{1}^{2}{\alpha}_{2},$ ${\beta}_{2}\left(\alpha \right)=-{\alpha}_{1}^{2}-{\alpha}_{1}{\alpha}_{2}-{\alpha}_{2}^{2}$, $({\alpha}_{1},{\alpha}_{2})=(0.9,-0.9)$ and $({\rho}_{1},{\phi}_{1})=(0.183,0)$ (red orbit), $({\rho}_{1},{\phi}_{1})=(0.16,0)$ (blue orbit), $({\rho}_{1},{\phi}_{1})=(0.14,0)$, (green orbit), respectively.

**Figure 17.**Numerical simulations for the map (10) when ${\beta}_{1}\left(\alpha \right)=2{\alpha}_{1}^{3}+{\alpha}_{2}^{3}+{\alpha}_{1}^{2}{\alpha}_{2}$, ${\beta}_{2}\left(\alpha \right)=-{\alpha}_{1}^{2}-{\alpha}_{1}{\alpha}_{2}-{\alpha}_{2}^{2}$: (

**a**) when $({\alpha}_{1},{\alpha}_{2})=(0.1,0.1)$, three orbits having $({\rho}_{1},{\phi}_{1})=(0.2,0)$ (red color), $({\rho}_{1},{\phi}_{1})=(0.16,0)$ (blue color), and $({\rho}_{1},{\phi}_{1})=(0.11,0)$ (green color) are given, corresponding this case to region 2 from Figure 11a; (

**b**) when $({\alpha}_{1},{\alpha}_{2})=(0.1,-0.112)$ and the three starting points of the orbits correspond to $({\rho}_{1},{\phi}_{1})=(0.2,0)$ (red orbit), $({\rho}_{1},{\phi}_{1})=(0.1711,0)$ (blue orbit), $({\rho}_{1},{\phi}_{1})=(0.11,0)$, (green orbit), respectively, we obtain the phase portrait corresponding to region 1 from Figure 11a.

**Table 1.**The sign of ${\beta}_{1}\left(T\right)$ when there are three roots ${e}_{1},\phantom{\rule{0.277778em}{0ex}}{e}_{2},\phantom{\rule{0.277778em}{0ex}}{e}_{3}$.

T | $(-\mathit{\infty},{\mathit{e}}_{1})$ | ${\mathit{e}}_{1}$ | $({\mathit{e}}_{1},{\mathit{e}}_{2})$ | ${\mathit{e}}_{2}$ | $({\mathit{e}}_{2},{\mathit{e}}_{3})$ | ${\mathit{e}}_{3}$ | $({\mathit{e}}_{2},\mathit{\infty})$ |
---|---|---|---|---|---|---|---|

sign${\beta}_{1}\left(T\right)$ | sign($-a$) | 0 | sign(a) | 0 | sign(−a) | 0 | sign(a) |

T | $(-\mathit{\infty},{\mathit{e}}_{1})$ | ${\mathit{e}}_{1}$ | $({\mathit{e}}_{1},\mathit{\infty})$ |
---|---|---|---|

sign${\beta}_{1}\left(T\right)$ | sign(−a) | 0 | sign(a) |

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**MDPI and ACS Style**

Lugojan, S.; Ciurdariu, L.; Grecu, E.
Chenciner Bifurcation Presenting a Further Degree of Degeneration. *Mathematics* **2022**, *10*, 1603.
https://doi.org/10.3390/math10091603

**AMA Style**

Lugojan S, Ciurdariu L, Grecu E.
Chenciner Bifurcation Presenting a Further Degree of Degeneration. *Mathematics*. 2022; 10(9):1603.
https://doi.org/10.3390/math10091603

**Chicago/Turabian Style**

Lugojan, Sorin, Loredana Ciurdariu, and Eugenia Grecu.
2022. "Chenciner Bifurcation Presenting a Further Degree of Degeneration" *Mathematics* 10, no. 9: 1603.
https://doi.org/10.3390/math10091603