On the number of limit cycles of a class of polynomial differential systems

Abstract

We study the number of limit cycles of polynomial differential systems of the forṁwhere g 1 , f 1 , g 2 and f 2 are polynomials of a given degree. Note that when g 1 (x) = f 1 (x) = 0, we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centreẋ = y,ẏ = −x using the averaging theory of first and second order.

Limit Cycles for a Class of Polynomial Differential Systems Via Averaging Theory

“…Proof. The integral A i,j (2π) can be calculated using the integrals (11), (9), (12) and (10) of the of appendix.…”

“…In [12] the authors use the averaging theory of first and second order to study the system {ẋ = y − ε(g 11 (x) + f 11 (x) y) − ε 2 (g 12 (x) + f 12 (x) y), y = −x − ε(g 21 (x) + f 21 (x) y) − ε 2 (g 22 (x) + f 22 (x) y),…”

On the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems

“…This problem restricted to continuous planar polynomial differential systems is the well known Hilbert's 16th problem, see for example [Li, 2003]. Up to now, there have been many achievements concerning the existence, uniqueness and the number of limit cycles, see for example [García et al, 2014;Justino & Jorge, 2012;Li & Llibre, 2012;Llibre, 2010;Llibre & Mereu, 2013Llibre et al, 2015;Llibre & Valls, 2012, 2013a, 2013bLloyd & Lynch, 1988;Martins & Mereu, 2014;Shen & Han, 2013;Sun, 1992;Xiong & Zhong, 2013] and references therein. A limit cycle bifurcating from a single degenerate singular point is called a small amplitude limit cycle, and the one bifurcating from periodic orbits of a linear center is called a medium amplitude limit cycle.…”

“…A limit cycle bifurcating from a single degenerate singular point is called a small amplitude limit cycle, and the one bifurcating from periodic orbits of a linear center is called a medium amplitude limit cycle. In [Llibre & Valls, 2012], the authors studied the number F. Jiang et al of medium amplitude limit cycles for a class of polynomial differential systems of the form…”

Restricted independence in displacement function for better estimation of cyclicity

“…As it appears in the books [3,5,19] and articles [4,6,13,14,18,20], the discussion on the center-focus equilibria is one of the most important problems in ordinary differential equations. A center-focus equilibrium is an equilibrium at which the linear part of the differential system has a pair of nonvanished pure imaginary eigenvalues.…”