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arXiv · 2303.14403

Centers and invariant straight lines of planar real polynomial vector fields and its configurations

Hongjin He·Changjian Liu·Dongmei Xiao
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Abstract

In the paper, we first give the least upper bound formula on the number of centers of planar real polynomial Hamiltonian vector fields. This formula reveals that the greater the number of invariant straight lines of the vector field and the less the number of its centers. Then we obtain some rules on the configurations of centers of planar real polynomial Hamiltonian Kolmogorov vector fields when the number of centers is exactly the least upper bound. As an application of these results, we give an affirmative answer to a conjecture on the topological classification of configurations for the cubic Hamiltonian Kolmogorov vector fields with four centers. Moreover, we discuss the relationship between the number of centers of planar real polynomial vector fields and the existence of limit cycles, and prove that cubic real polynomial Kolmogorov vector fields have no limit cycles if the number of its centers reaches the maximum. More precisely, it is shown that the cubic real polynomial Kolmogorov vector field must have an elementary first integral in $\mathbb{R}^2\setminus\{xy=0\}$ if it has four centers, and the number of configurations of its centers is one more than that of the cubic polynomial Hamiltonian Kolmogorov vector fields.

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Hongjin He, Changjian Liu, Dongmei Xiao. 2023-03-25. Centers and invariant straight lines of planar real polynomial vector fields and its configurations. https://doi.org/10.3934/dcdsb.2023144 10.1016/j.bulsci.2026.103834

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