The number of limit cycles of a quintic polynomial system

Abstract

a b s t r a c tIn this paper we consider the bifurcation of limit cycles of the systemẋ = y(for ε sufficiently small, where a, b ∈ R − {0}, and P, Q are polynomials of degree n, we obtain that up to first order in ε the upper bounds for the number of limit cycles that bifurcate from the period annulus of the quintic center given by ε = 0 are (3/2)(n + sin 2 (nπ /2)) + 1 if a = b and n − 1 if a = b. Moreover, there are systems with at least (3/2)(n + sin 2 (nπ /2)) + 1 if a = b and, n − 1 limit cycles if a = b.

Upper bounds for the number of zeroes for some Abelian integrals

“…The special case with four lines given by G(x, y) = (x 2 − a 2 )(y 2 − b 2 ) was studied in [1]. When a = b the authors showed that Z(I) ≤ 3n+5 2 , n is odd, 3n+2 2 , n is even, and it was claimed that these bounds are sharp.…”

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.…”

Limit cycles bifurcating from a perturbed quartic center

“…Basically, four methods have been used to perform such studies and they are based on: the Poincaré return map (see for instance [7,8,18]), the Poincaré-Pontrjagin-Melnikov integrals or Abelian integrals that are equivalent in the plane (see [2,11,3,4,6,10,25,26]), the inverse integrating factor (see [12,13,14,24]), and the averaging method which in the plane is also equivalent to the Abelian integrals (see for instance [5,17,19]). …”

“…Using the averaging theory of first order in [6], and the Melnikov method in [3], the authors give upper and lower bounds for the maximum number of limit cycles bifurcating from the period annulus of a cubic or quintic center, respectively. The lower bounds in both cases are 3[(n − 1)/2] + 2 and 3[(n + 1)/2] + 1, respectively.…”

“…For doing this study we transform the differential system (2) in the equivalent differential equation dr dθ = ε P n (r cos θ, r sin θ) cos θ + Q n (r cos θ, r sin θ) sin θ (r cos θ + a)(r sin θ + b)(r cos θ + c) + ε 2 G(θ, r, ε), (3) by using the change to polar coordinates x = r cos θ, y = r sin θ and taking θ as the new independent variable of the differential system. System (3) is defined on the disc r < d, with d = min{|a|, |b|, |c|} and being G a 2π-periodic function of θ.…”