The maximum number of centers for planar polynomial Kolmogorov differential systems. (English) Zbl 1542.34026
The authors provide interesting results in one of the 33 problems proposed by A. Gasull [S\(\vec{\text{e}}\)MA J. 78, No. 3, 233–269 (2021; Zbl 1487.37024)], more precisely, Problem 10: Determine the maximum number of centers for planar polynomial differential systems of degree \(n \geq 4\). A. Cima et al. [Proc. Am. Math. Soc. 118, No. 1, 151–163 (1993; Zbl 0787.34030)], proved that \( \left[ \frac{n^2 + 1}{2} \right] \leq C(n) \leq \frac{n^2 +n}{2} - 1, \ \forall n \geq 2\), where \(C(n)\) represents the maximum number of centers for a class of polynomials systems and \([\cdot]\) is the integer part of the number.
The paper considers the above mentioned Problem 10 for a class of polynomial systems of the form \[ \dot{x}=x P(x,y), \ \ \dot{y}=y Q(x,y),\tag{1} \] where \(P(x, y)\) and \(Q(x, y)\) are real polynomials of degree \(n - 1\) and \(m -1\), respectively and \(n \geq m\). The authors call such systems as planar polynomial Kolmogorov differential systems of degree n.
The main results of the paper are Theorem 1.1 and Theorem 1.2. Theorem 1.1 provides lower and upper bounds for the maximum number of centers for system (1) of degree \(n\), more precisely it is proved that \[ \left[ \frac{n^2 - 1}{2} \right] \leq K_C(n) \leq \frac{n^2 +n}{2} - 2, \ \forall n \geq 2, \] where \(K_C(n)\) denotes the maximum number of centers for system (1) of degree \(n\). Theorem 1.2 states that \(K_C(4)=7\), that is, the maximum number of centers for a quartic system of the form (1) is seven.
Furthermore, the authors propose the following conjecture: \(K_C(n) = \left[ \frac{n^2 - 1}{2}\right]\) \(\forall n \in \mathbb{N}\).
The paper considers the above mentioned Problem 10 for a class of polynomial systems of the form \[ \dot{x}=x P(x,y), \ \ \dot{y}=y Q(x,y),\tag{1} \] where \(P(x, y)\) and \(Q(x, y)\) are real polynomials of degree \(n - 1\) and \(m -1\), respectively and \(n \geq m\). The authors call such systems as planar polynomial Kolmogorov differential systems of degree n.
The main results of the paper are Theorem 1.1 and Theorem 1.2. Theorem 1.1 provides lower and upper bounds for the maximum number of centers for system (1) of degree \(n\), more precisely it is proved that \[ \left[ \frac{n^2 - 1}{2} \right] \leq K_C(n) \leq \frac{n^2 +n}{2} - 2, \ \forall n \geq 2, \] where \(K_C(n)\) denotes the maximum number of centers for system (1) of degree \(n\). Theorem 1.2 states that \(K_C(4)=7\), that is, the maximum number of centers for a quartic system of the form (1) is seven.
Furthermore, the authors propose the following conjecture: \(K_C(n) = \left[ \frac{n^2 - 1}{2}\right]\) \(\forall n \in \mathbb{N}\).
Reviewer: Wilker Fernandes (São João del-Rei)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C08 | Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) |
| 34C25 | Periodic solutions to ordinary differential equations |
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