Linear Estimation of the Number of Zeros of Abelian Integrals for Some Cubic Isochronous Centers

Abstract

This paper consists of two parts. In the first part we study the relationship between conic centers (all orbits near a singular point of center type are conics) and isochronous centers of polynomial systems. In the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics, when we perturb such systems inside the class of all polynomial systems of degree n. © 2002 Elsevier Science (USA)

Hilbert's 16th Problem and Bifurcations of Planar Polynomial Vector Fields

“…With the help of numerical analysis (using Maple or Mathematica) we showed that there exist parameter groups such that at least 15 to 24 limit cycles exist, having the configurations of compound eyes in these systems. The cases for q = 2 and 3 are being reported separately in [Chan et al, 2001] and [Li et al, 2002] where at least 15 and 23 limit cycles are respectively obtained. The case for q = 6 is discussed in [Li et al, 2002].…”

Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2

“…Thus, we get the systemẋ = 2y(x + α) 2 , y = −2(x + α)(αx − y 2 ), with α ̸ = 0. In [12] the authors studied the cubic polynomial differential systems having a rational first integral of degree 2 whose phase portraits correspond to the phase portraits P 1 , P 3 and P 4 of Figure 1. These systems were denoted in [12] by (A), (B) and (C).…”

“…In [12] the authors studied the cubic polynomial differential systems having a rational first integral of degree 2 whose phase portraits correspond to the phase portraits P 1 , P 3 and P 4 of Figure 1. These systems were denoted in [12] by (A), (B) and (C). They also proved that all the centers of these systems are reversible and isochronous, see [12, p. 314].…”

“…The second column presents the upper bounds M provided in [12], and the third column presents the maximum number reached N of limit cycles bifurcating from the periodic orbits surrounding the systems P k when they are perturbed inside the class of all cubic polynomial differential systems using the averaging theory of first order. The fourth column presents the maximum number reached I of infinitesimal limit cycles bifurcating from origin for systems P k using the averaging theory of fifth order.…”

Limit Cycles for a Class of Continuous and Discontinuous Cubic Polynomial Differential Systems

“…In this paper we shall perturb isochronous centers. The perturbation of some of these centers have already been studied see for instance [7,14]. For a survey on isochronous centers see [6].…”