# Symmetries in Phase Portrait

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Equilibrium Points Set Stratification

**Example**

**1.**

**Example**

**2.**

## 3. Basic Facts

**Definition**

**2.**

#### 3.1. Euler–Jacobi Formula

**Theorem**

**1**

#### 3.2. Application of the Euler–Jacobi Formula

**Corollary**

**1.**

**Corollary**

**2.**

**Theorem**

**2.**

**Proof.**

#### 3.3. The Scope of the Euler–Jacobi Formula

**Definition**

**3.**

**Proposition**

**1.**

**Proof.**

- Nonisolated equilibrium points do not make sense for summation in (9).
- Zero determinant in the denominator does not allow division.

#### 3.4. Generalized Euler-Jacobi Formula

**Proposition**

**2**

**Theorem**

**3.**

**Remark**

**1.**

## 4. Symmetry Constructions

**Definition**

**4.**

#### 4.1. Matrices with the Same Spectrum

**Definition**

**5.**

- (a)
- Two matrices A and B are simultaneously diagonalizable then they commute $AB=BA$,
- (b)
- Not every pair of commuting matrices are simultaneously diagonalizable.
- (c)
- Two diagonalizable matrices preserves eachother’s eigenspaces.
- (d)
- The set of simultaneously diagonalizable matrices generate a toral Lie algebra

#### 4.2. Circulant Matrices

**Definition**

**6.**

#### 4.3. Diagonalization of Circulant Matrices

#### 4.4. Circulant Convolution

**Definition**

**7.**

**Definition**

**8.**

**Theorem**

**4.**

**Proof.**

**Example**

**3.**

**Example**

**4.**

## 5. Conclusions

- Our main goal is to create examples of ODEs that have a high symmetry.
- We show the necessary conditions for homogeneous ODEs with high symmetry to have charpoly of the Jacobian matrix of form ${\lambda}^{n}-{k}^{n}$ (see Theorem 2).
- We explicitly indicate the connection of the circulant matrices theory on possible symmetries in phase portraits. Namely, the system ${x}^{\prime}=D{x}^{\odot k}-x$ written using circulant convolution notation, has a high symmetry in its phase portrait. (See Theorem 4 and it graphical representation in Appendix A)

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Symmetries in the Plane Phase Portrait

**Example**

**A1.**

**Example**

**A2.**

## Appendix B. Basic Definitions

- We consider an autonomous system ${x}^{\prime}=P\left(x\right)$.
- An equilibrium (singular) point ${x}^{*}$ of an ODE ${x}^{\prime}\left(t\right)=P\left(x\left(t\right)\right)$ is one where a solution of the differential equation remains fixed at ${x}^{*}$ for all time.
- The solutions to the linearized system near an equilibrium point ${x}^{*}$ are approximately closed to the solutions of the actual system provided that ${x}^{*}$ is hyperbolic.
- A diffeomorphism is an invertible map such that both the mapping and its inverse are continuously differentiable.
- The definition of the geometrical equivalence can be generalized to cover more general cases when the state space is a complete metric or, in particular, is a Banach space.
- The phase portraits of topologically equivalent systems are often also called topologically equivalent.

## References

- Belov, A.; Bokut, L.; Rowen, L.; Yu, J.T. The Jacobian Conjecture, Together with Specht and Burnside-Type Problems. In Automorphisms in Birational and Affine Geometry; Springer: Cham, Switzerland, 2014; Volume 79, pp. 249–285. [Google Scholar]
- Davis, P.J. Circulant Matrices; Wiley: New York, NY, USA; Chichester, UK; Brisbane, Australia, 1979. [Google Scholar]
- Kra, I.; Simanca, S.R. On circulant matrices. Not. Am. Math. Soc.
**2012**, 59, 368–377. [Google Scholar] [CrossRef] - Grobman, D. Homeomorphisms of Systems of Differential Equations. Dokl. Akad. Nauk
**1959**, 128, 880–881. (In Russian) [Google Scholar] - Hartman, P. A Lemma in the Theory of Structural Stability of Differential Equations. Proc. Am. Math. Soc.
**1960**, 11, 610–620. [Google Scholar] [CrossRef] - Arnold, V.I.; Varchenko, A.N.; Gusein-Zade, S.M. Singularities of Differentiable Maps; Birkhäuser: Boston, MA, USA, 1985; Volume I. [Google Scholar]
- Balanov, Z.; Kononovich, A.; Krasnov, Y. Projective dynamics of homogeneous systems: Local invariants, syzygies, and the global residue theorem. Proc. Edinb. Math. Soc. Ser. II
**2012**, 55, 577–589. [Google Scholar] [CrossRef] - Arnold, V.I. Geometric Methods in the Theory of Ordinary Differential Equations, 2nd ed.; Springer: New York, NY, USA, 1998. [Google Scholar]
- Krasnov, Y.; Tkachev, V.G. Idempotent geometry in generic algebras. Adv. Appl. Clifford Algebr.
**2018**, 28, 84–98. [Google Scholar] [CrossRef] [Green Version] - Griffiths, P.; Harris, J. Principles of Algebraic Geometry; Wiley-Interscience: New York, NY, USA, 1978. [Google Scholar]
- Balanov, Z.; Krasnov, Y. On good deformations of A
_{m}-singularities. Discret. Cont. Dyn. Syst. Ser. S**2019**, 12, 1851–1866. [Google Scholar] - Krasnov, Y. Spectral Theory for Nonlinear Operators: Quadratic Case. In Modern Methods in Operator Theory and Harmonic Analysis; Springer: Cham, Switzerland, 2019; Volume 291, pp. 199–216. [Google Scholar]

**Figure 1.**${D}_{3}$ symmetry stratification in phase portrait of (1) near ${x}_{1}^{*},{x}_{2}^{*},{x}_{3}^{*}$.

**Figure 2.**Phase portraits of ${P}_{1},{P}_{2}$ in (

**a**,

**b**) follow the same set of a hyperbolic equilibrium points in (

**c**).

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**MDPI and ACS Style**

Krasnov, Y.; Koylyshov, U.K.
Symmetries in Phase Portrait. *Symmetry* **2020**, *12*, 1123.
https://doi.org/10.3390/sym12071123

**AMA Style**

Krasnov Y, Koylyshov UK.
Symmetries in Phase Portrait. *Symmetry*. 2020; 12(7):1123.
https://doi.org/10.3390/sym12071123

**Chicago/Turabian Style**

Krasnov, Yakov, and Umbetkul K. Koylyshov.
2020. "Symmetries in Phase Portrait" *Symmetry* 12, no. 7: 1123.
https://doi.org/10.3390/sym12071123