Limit cycles bifurcating from a perturbed quartic center

Abstract

Abstract. We consider the quartic centerẋ = −yf (x, y),ẏ = xf (x, y), with f (x, y) = (x+a)(y+b)(x+c) and abc = 0. Here we study the maximum number σ of limit cycles which can bifurcate from the periodic orbits of this quartic center when we perturb it inside the class of polynomial vector fields of degree n, using the averaging theory of first order. We prove that 4[(n − 1)/2] + 4 ≤ σ ≤ 5[(n − 1)/2 + 14, where [η] denotes the integer part function of η.

Upper bounds for the number of zeroes for some Abelian integrals

“…The proof of Theorem 1.2 will use the known as Derivation-Division procedure (see for instance [11, p. 119]). First we give two technical results useful to compute successive derivatives that appear applying this procedure to the function (5). The proof of the first one is very simple and implies that, in general,…”

“…Several previous works handle this problem for different particular choices of small values of K 1 and K 2 . The following cases have been studied: one line in [10]; two parallel lines in [13]; two orthogonal lines in [2]; three lines, two of them parallel and one perpendicular in [5]; and four lines with a special configuration in [1]. Other related works are [8], with G(x, y) any quadratic polynomial; [12], one multiple singular line; or [7] for K isolated singular points.…”

Limit Cycles Bifurcating from a Family of Reversible Quadratic Centers via Averaging Theory

“…The tools for studying the limit cycles bifurcating from the periodic orbits surrounding a center are the Poincaré return map (see for instance [3,20]), the Poincaré-Melnikov integrals (see for example [13,14]), the Abelian integrals (see [1,6,28]), the averaging theory (see for instance [2,7]), and the inverse integrating factor (see [11]). While the first three methods only provide the number of bifurcated limit cycles, the averaging method and method which use the inverse integrating factor can also give the shape of bifurcated limit cycles.…”

Poincaré Bifurcations of Two Classes of Polynomial Systems

“…In [12], the authors studied the number of limit cycles of system (1) with ( , ) = ( + )( + ) and obtain that the system can have at most 3[( − 1)/2] + 2 limit cycles if ̸ = and 2[( − 1)/2] + 1 if = , respectively. In [13], the authors studied the case the curves ( , ) = 0 are three lines, two of them parallel and one perpendicular, and [14,15] studied the case the curves are ( > 3) lines, and any two of them are parallel or perpendicular directions. The authors in [16] studied the case the curves are consistent by nonzero points.…”