Nondegenerate centers for Abel polynomial differential equations of second kind

Abstract

In this paper we study the center problem for Abel polynomial differential equations of second kind. Computing the focal values and using modular arithmetics and Gröbner bases we find the center conditions for such systems for lower degrees.

Centers of planar generalized Abel equations

“…In fact the Abel equations are of the form ṙ = a(θ) + b(θ)r + c(θ)r 2 + d(θ)r 3 , and they appeared by the first time in the works of Niels Henryk Abel, see [7]. Today there are more than 1400 papers in MathSciNet with the name "Abel equation" in their tittle, see for instance the papers [1,2,3,5,6,8] for results on centers in the Abel equations and the references quoted therein.…”

Transformation Method for Generating Periodic Solutions of Abel’s Differential Equation

“…plays an important role in many physical and technical applications [1,2]. The mathematical properties of Equation (1) have been intensively investigated in the mathematical and physical literature [3][4][5][6][7][8][9][10][11]. S. S. Misthry, S. D. Maharaj, and P. G. L. Leach [12] introduced a new transformation at the boundary that leads to an Abel's equation and showed explicitly that a variety of exact solutions can be generated from the Abel equation.…”

The Fixed Point Theory and the Existence of the Periodic Solution on a Nonlinear Differential Equation

“…Recently, A. Cima, A. Gasull, and F. Manosas [8] gave the maximum number of polynomial solutions of some integrable polynomial Abel differential equations; Jaume Giné Claudia and Valls [9] studied the center problem for Abel polynomial differential equations of second kind; Jianfeng Huang and Haihua Liang [10] were devoted to the investigation of Abel equation by means of Lagrange interpolation formula; they gave a criterion to estimate the number of limit cycles of the Abel's equations; Berna Bülbül and Mehmet Sezer [11] introduced a numerical power series algorithm which is based on the improved Taylor matrix method for the approximate solution of Abel-type differential equations; Ni et al [12] discussed the existence and stability of the periodic solutions of (1) and obtained the sufficient conditions which guaranteed the existence and stability of the periodic solutions for (1) from a particular one.…”