Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems
Abstract
:1. Introduction
2. Autonomous Polynomial Dynamical Systems
2.1. Classification of Autonomous Polynomial Dynamical Systems
2.2. On Integrability of Polynomial Dynamical Systems
2.3. Symmetries of Quadratic Polynomial DSs
2.4. Extension to Higher-Degree Polynomials
3. Classes D of Solutions to Quadratic Polynomial DSs. Discriminant Criterion. Linearization
3.1. Discriminant Criterion
- (1)
- D: and
- (2)
- D: and
- (3)
- D: and
- (4)
- D: and
3.2. Linearization in D
3.3. Analysis of Solutions to Cauchy Problems
- at any increases at and decreases at (black lines in Figure 3);
- at any decreases at increases at, and is situated in the band between and (blue lines in Figure 3);
- for increases at, decreases at , and is situated outside (green lines in Figure 3);
- at any decreases at increases at , and is situated in (magenta lines in Figure 3);
- at any decreases at increases at , and is situated outside (red lines in Figure 3);
- at increases at decreases at , and is situated in (orange lines in Figure 3).
- (a)
- for and any the solutions are monotonically increasing functions that lie in the strip between two special stationary solutions and
- (b)
- for or and any the solutions are monotonically decreasing unbounded functions that lie outside the strip between two stationary solutions. In the considered case, the entire phase plane is filled with the solution curves.
4. Qualitative Theory of Polynomial Quadratic DSs. Visualization of Transitions
4.1. Analysis of Solutions and Phase Portraits
4.2. Atlas of Phase Portraits
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
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(a) | (b) |
Class | Example of Equation | General Solutions of | |
---|---|---|---|
1 | D | , | |
2 | D | ||
3 | D | ||
4 | D | ||
5 | D | ||
6 | D | ||
7 | D | , , | |
8 | D | , | |
9 | D |
Stable Node | Unstable Node | Saddle | |
---|---|---|---|
Stable Node | |||
Unstable Node | |||
Saddle |
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Shestopalov, Y.; Shakhverdiev, A. Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems. Symmetry 2021, 13, 1884. https://doi.org/10.3390/sym13101884
Shestopalov Y, Shakhverdiev A. Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems. Symmetry. 2021; 13(10):1884. https://doi.org/10.3390/sym13101884
Chicago/Turabian StyleShestopalov, Yury, and Azizaga Shakhverdiev. 2021. "Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems" Symmetry 13, no. 10: 1884. https://doi.org/10.3390/sym13101884
APA StyleShestopalov, Y., & Shakhverdiev, A. (2021). Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems. Symmetry, 13(10), 1884. https://doi.org/10.3390/sym13101884