Center conditions for generalized polynomial Kukles systems

Abstract

In this paper we study the center problem for certain generalized Kukles systemṡ x = y,ẏ = P 0 (x) + P 1 (x)y + P 2 (x)y 2 + P 3 (x)y 3 , where P i (x) are polynomials of degree n, P 0 (0) = 0 and P 0 (0) < 0. Computing the focal values and using modular arithmetics and Gröbner bases we find the center conditions for such systems when P 0 is of degree 2 and P i for i = 1, 2, 3 are of degree 3 without constant terms. We also establish a conjecture about the center conditions for such systems.

Nondegenerate centers for Abel polynomial differential equations of second kind

“…Note that 3+e 1 y = 0 is an invariant algebraic curve of system (25). However with this curve and doing the change y = φ(x)z/(z+1), where φ(x) = −3/e 1 the transformed system is not more simple than the previous one as happens in the generalized Kukles systems, see [14]. Nevertheless system (25) has the invariant curves…”

“…More precisely, we want to find the different type of centers that appear for such systems and we aim to know if it is true that all the types of centers that appear at the origin of system (5) are algebraic reducible or Liouville integrable, see [16]. However, this goal is too ambitious because within these systems there are the Liénard systems [8,13], the Cherkas systems [9,16], the Kukles systems [21,26] and the generalized Kukles systems [14,22].…”

“…System 7with f 2 = 0 is a generalized Kukles system, see [14]. This case was studied in [14] with the additional restriction c 0 = d 0 = 0.…”

“…System 7with f 2 = 0 is a generalized Kukles system, see [14]. This case was studied in [14] with the additional restriction c 0 = d 0 = 0. The first goal of this paper is to study system (7) with f 2 = 0 and arbitrary values of c 0 and d 0 , see Theorem 1 below.…”

“…However, for completeness, we also discuss this particular case here. System (14) has the particular solutions f 1 = 1 + e 1 x and…”