# Bifurcations Associated with Three-Phase Polynomial Dynamical Systems and Complete Description of Symmetry Relations Using Discriminant Criterion

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## Abstract

**:**

## 1. Introduction

## 2. A Family of Polynomial Dynamical Systems

#### 2.1. Matrix Representation of Polynomial Dynamical Systems

#### 2.2. Discriminant Criterion and Matrix Representation of 3D Polynomial Dynamical Systems

#### 2.3. Symmetry Relations on the Sets of Coefficient Matrices and D-Vectors

## 3. Classification of Solutions to Autonomous Polynomial Equations

#### 3.1. Representations of Autonomous and Integrable Polynomial Dynamical Systems

#### 3.2. General Solutions to Autonomous Second-Order Polynomial Equations

**(a)**$a=0$; $x\left(t\right)={\frac{1}{b}(-c+Ce}^{t}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}b\ne 0.$In what follows, we will consider the equations with $a\ne 0.$**(b)**D > 0; there are three solution families$${\phantom{\rule{4pt}{0ex}}x\left(t;C\right)=x}_{1}+\phantom{\rule{4pt}{0ex}}\frac{\sqrt{D}}{a}\frac{1}{-1+C{e}^{-\sqrt{D}t}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\phantom{\rule{4pt}{0ex}}x}_{1,2}=-{x}_{0}\pm \frac{\sqrt{D}}{2a},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\phantom{\rule{4pt}{0ex}}x}_{0}=\frac{b}{2a},\phantom{\rule{4pt}{0ex}}D={b}^{2}-4ac,$$Family U, with $C>0$: solutions $x\left(t;C\right)={x}_{\mathrm{U}}\left(t;C\right)$ are not stable, since there is a “movable” singular point with $t={t}^{*}\left(C\right)=\frac{lnC\phantom{\rule{4pt}{0ex}}}{\sqrt{D}}$; next, they are (i) monotonically increasing because $\frac{dx\left(t;C\right)}{dt}=\sqrt{D}\frac{C{e}^{-\sqrt{D}t}}{{(C{e}^{-\sqrt{D}t}-1)}^{2}}>0$ for C > 0; (ii) satisfy the condition $x\left(t;C\right)>{x}_{2},\phantom{\rule{4pt}{0ex}}x<{t}^{*},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}x\left(t;C\right)<{x}_{1},\phantom{\rule{4pt}{0ex}}x>{t}^{*}$; and (iv) have two horizontal asymptotes $x={x}_{1}\phantom{\rule{4pt}{0ex}}\mathrm{or}\phantom{\rule{4pt}{0ex}}x={x}_{2}$.Family S, with $C>0$: solutions $x\left(t;C\right)={x}_{\mathrm{S}}\left(t;C\right)$ are stable and ${x}_{1}<x\left(t;C\right)<{x}_{2}$.Family T, : $x={x}_{1}$ and $x={x}_{2}$ are time-independent solutions such that $x\left(0\right)={x}_{1}\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}x\left(0\right)={x}_{2},$ and the first corresponds to $C=0$ in $\phantom{\rule{4pt}{0ex}}x\left(t;C\right)=x}_{1}+\phantom{\rule{4pt}{0ex}}\frac{\sqrt{D}}{a}\frac{1}{-1+C{e}^{-\sqrt{D}t}$. These stationary solutions $x={x}_{1}$ and $x={x}_{2}$ are ’nonisolated’: in every neighborhood, there is an infinite number of ‘regular’ solutions ${x}_{\mathrm{U}}\left(t;C\right)$ or ${x}_{\mathrm{S}}\left(t;C\right)$.**(c)**, $D=0$: all the corresponding solutions$$\phantom{\rule{4pt}{0ex}}x\left(t;C\right)=-\frac{1}{a}\left(\frac{b}{2}+\frac{1}{t-C}\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$$**(d)**, $D<0$: the corresponding solutions$$x\left(t;C\right)=-{x}_{0}+\frac{\sqrt{-D}}{2a}tan\frac{\sqrt{-D}}{2}\phantom{\rule{4pt}{0ex}}\left(t+C\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}D<0,$$

#### 3.3. Equivalence Classes of D-Vectors and General Solutions to Autonomous Polynomial Equation Systems

#### 3.4. Description of All Possible Solution Combinations in Terms of Discriminants

#### 3.5. Analysis of Solutions to Cauchy Problems

## 4. Analysis of Bifurcations

## 5. Conclusions

- -
- To develop the method of S- and D-vectors and the discriminant criterion to the polynomial DSs of higher dimensions and the order of the involved polynomials.
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- To clarify that the only type of bifurcations that may occur in quadratic polynomial DSs investigated in this paper is the discovered ‘twisted fold’.
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- To investigate the relations between the described symmetries of the D- and S-vectors and the possible symmetries of solutions to the polynomial DSs.
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- To find the symmetry-breaking bifurcations characteristic to the polynomial DSs under study.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Autonomous Polynomial Dynamical Systems Integrable in Elementary Functions

**Lemma A1.**

**Proof.**

**Proof.**

#### Appendix A.2. Examples with Bifurcations

## References

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**Figure 1.**Surface ${\mathbf{X}}_{{D}_{1}}^{-0-}\cup {\mathbf{X}}_{{D}_{1}}^{+0-}$ formed by the parametrized curves $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{-0-}\phantom{\rule{4pt}{0ex}}$ (black), ${C}_{1}=-1$, $-4<{D}_{1}<0$, and $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{+0-},0\le t\le 1.3$ (green), $0<{D}_{1}<4$, with ${a}_{11}={a}_{13}={a}_{21}=1$ and ${a}_{22}=2$ given by (43)–(46) corresponding to the solutions from the sets D${}^{-0-}$ and D${}^{+0-}$. The explicit formulas (A22)–(A26) for the parametrized components are in Appendix A.

**Figure 2.**Surface ${\mathbf{X}}_{{D}_{1}}^{-0-}\cup {\mathbf{X}}_{{D}_{1}}^{+0-}$ formed by the parametrized curves $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{-0-}$ (right, black), $-4<{D}_{1}<0$, and $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{+0-},\phantom{\rule{4pt}{0ex}}0\le t\le 1.3$ (left, blue), $0<{D}_{1}<4$, with ${a}_{11}={a}_{13}={a}_{21}=1$ and ${a}_{22}=2$ given by (43)–(46) corresponding to the solutions from the sets D${}^{-0-}$ and D${}^{+0-}$. The explicit formulas (A16)–(A21) for the parametrized components are in Appendix A.

**Figure 3.**Surface ${\mathbf{X}}_{{D}_{1}}^{-0-}\cup {\mathbf{X}}_{{D}_{1}}^{+0-}$ formed by the parametrized curves $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{-0-}\phantom{\rule{4pt}{0ex}}$ (black), ${C}_{1}=-1.3$, $-4<{D}_{1}<0$, and $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{+0-}\phantom{\rule{4pt}{0ex}},0\le t\le 1.3$ (green), $0<{D}_{1}<4$, with ${a}_{11}={a}_{13}={a}_{21}=1$ and ${a}_{22}=2$ given by (43)–(46) corresponding to the solutions from the sets D${}^{-0-}$ and D${}^{+0-}$.

**Figure 4.**Surface ${\mathbf{X}}_{{D}_{1}}^{0--}\cup {\mathbf{X}}_{{D}_{1}}^{+0-}$ formed by the parametrized curves $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{\mathbf{0}--}\phantom{\rule{4pt}{0ex}}$ (black), ${C}_{1}=-1$, $-4<{D}_{1}<0$, and $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{+0-}\phantom{\rule{4pt}{0ex}},0\le t\le 1.3$ (green), $0<{D}_{1}<4$, with ${a}_{11}={a}_{13}={a}_{21}=1$ and ${a}_{22}=2$ given by (43)–(46) corresponding to the solutions from the sets D${}^{0--}$ and D${}^{+0-}$.

**Figure 5.**Surface ${\mathbf{X}}_{{D}_{1}}^{-0-}$ formed by the parametrized curves $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{-0-}\phantom{\rule{4pt}{0ex}}$, $0\le t\le 1.45,\phantom{\rule{4pt}{0ex}}-2\le {D}_{1}<0$, ${a}_{11}={a}_{13}=1$ given by (43)–(46) corresponding to the solutions from the set D${}^{-0-}$.

**Figure 6.**Enlarged fragment of the projection on the $(x,y)$-plane of the surface, presented in Figure 4, close to the point of bifuraction associated with the stationary solution $y=-1$ of the second component in system (51) (graphs of the parametrized curves $\mathbf{x}\left(t\right)={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{\mathbf{0}--}\phantom{\rule{4pt}{0ex}}$ (black), ${C}_{1}=-1$, $-4<{D}_{1}<0$, and $\mathbf{x}\left(t\right)={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{+0-}\phantom{\rule{4pt}{0ex}},0\le t\le 1.3$ (green), $0<{D}_{1}<4$, with ${a}_{11}={a}_{13}={a}_{21}=1$ and ${a}_{22}=2$ given by (43)–(46) corresponding to the solutions from the sets D${}^{0--}$ and D${}^{+0-}$).

**Figure 7.**Graphs of the parametrized curves $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{-0-}\phantom{\rule{4pt}{0ex}}$ (black), ${C}_{1}=-1$, ${C}_{2}=-0.04$, $-4<{D}_{1}<0$, and $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{+0-}\phantom{\rule{4pt}{0ex}},0\le t\le 1.4$ (yellow), $0<{D}_{1}<4$, with ${a}_{11}={a}_{13}={a}_{21}=1$ and ${a}_{22}=2$ given by (43)–(46) corresponding to the solutions from the sets D${}^{-0-}$ and D${}^{+0-}$.

**Figure 8.**Enlarged fragment of the surface close to the point of bifuraction associated with the stationary solution $y=-1$ of the second and third components in system (51) (graphs of the parametrized curves $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{-00}\phantom{\rule{4pt}{0ex}}$ (black), ${C}_{1}=-1$, $-4<{D}_{1}<0$, ${C}_{2}=1.6$, ${C}_{3}=2$, and $\mathbf{x}(t;{D}_{1})={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{+00}\phantom{\rule{4pt}{0ex}},0\le t\le 1.5$ (yellow), $0<{D}_{1}<4$, with ${a}_{11}={a}_{13}={a}_{21}=1$ and ${a}_{22}=2$ given by (43)–(46) corresponding to the solutions from the sets D${}^{-00}$ and D${}^{+00}$).

**Figure 9.**Graphs of the parametrized curves $\mathbf{x}\left(t\right)={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{---}\phantom{\rule{4pt}{0ex}}$, $0\le t\le 2,\phantom{\rule{4pt}{0ex}}-4\le {D}_{1},{D}_{3}<0$, ${a}_{11}={a}_{13}=1$, ${a}_{22}={a}_{32}=2$, ${C}_{j}=-1$, $j=1,2,3$, given by (43)–(46) corresponding to the solutions from the set D${}^{---}$ with two variable components.

**Figure 10.**Graphs of the parametrized curves $\mathbf{x}\left(t\right)={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{-+-}\phantom{\rule{4pt}{0ex}}$ (right) and $\mathbf{x}\left(t\right)={\mathbf{x}}_{\mathbf{1}}+{\mathbf{g}}^{++-}\phantom{\rule{4pt}{0ex}},0\le t\le 1.3,\phantom{\rule{4pt}{0ex}}-4\le {D}_{1}\le 6$, ${a}_{11}={a}_{13}=1$ given by (43)–(46) corresponding to the solutions from the sets D${}^{-+-}$ and D${}^{++-}$.

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**MDPI and ACS Style**

Shestopalov, Y.; Shakhverdiev, A.; Arefiev, S.V.
Bifurcations Associated with Three-Phase Polynomial Dynamical Systems and Complete Description of Symmetry Relations Using Discriminant Criterion. *Symmetry* **2024**, *16*, 14.
https://doi.org/10.3390/sym16010014

**AMA Style**

Shestopalov Y, Shakhverdiev A, Arefiev SV.
Bifurcations Associated with Three-Phase Polynomial Dynamical Systems and Complete Description of Symmetry Relations Using Discriminant Criterion. *Symmetry*. 2024; 16(1):14.
https://doi.org/10.3390/sym16010014

**Chicago/Turabian Style**

Shestopalov, Yury, Azizaga Shakhverdiev, and Sergey V. Arefiev.
2024. "Bifurcations Associated with Three-Phase Polynomial Dynamical Systems and Complete Description of Symmetry Relations Using Discriminant Criterion" *Symmetry* 16, no. 1: 14.
https://doi.org/10.3390/sym16010014