This paper provides a completed Melnikov analysis of any order for a class of perturbed polynomial differential systems, where the unperturbed system is a cubic center with a straight line of singular points. The emphasis on any order Melnikov function is crucial because of two major reasons: one is that higher order Melnikov functions are not in general expressible explicitly, the other one is the essential difficulty of computing elliptic integrals due to the complexity of many iterations. The exact upper bounds of the number of limit cycles bifurcated from the period annulus under different polynomial perturbations are obtained by analyzing the algebraic structures from explicit expressions of any order Melnikov functions. (c) 2024 Elsevier Inc. All rights reserved.

In this paper, the first fourth order Melnikov analyses are applied to find the limit cycles bifurcated from a cubic center under the perturbation of quadratic polynomials in ϵ up to the second order. The upper bounds of the number of limit cycles are given and can be reached.

In this paper, we consider a perturbation of planar general piecewise Hamiltonian systems with nonregular separation line. A general explicit expression of the first and second order Melnikov functions is presented for such piecewise Hamiltonian systems. In addition, the maximum number of limit cycles for piecewise linear differential system is an important topic that many researchers are concerned with. Different from considering a piecewise linear perturbation of a linear center in the previous research, we apply our explicit expression to estimate the number of limit cycles bifurcated from a class of piecewise linear Hamiltonian systems. More specifically, the unperturbed system we study is piecewise smooth and constructed by a linear center and a constant differential system, which is difficult to be considered by a polar coordinate transformation. And we obtain the upper bounds to be 4 and 5 for the limit cycles bifurcated from the periodic orbits by using the first and second order Melnikov functions, respectively. Moreover the upper bounds are sharp.

This article begins with a full description of the quadratic planar vector fields which display two centers. We follow the method proposed by Chengzhi Li and provide more detailed analysis of the different types of double centers using the classification: Hamiltonian, reversible, Lotka-Volterra, Q4, currently used for centers of quadratic planar vector fields. We also describe completely the different possible phase portraits and their Poincaré compactification. We show that the double center set is a semi-algebraic set for which we give an explicit stratification (see figure 2). Then we initiate a study of the perturbations within quadratic planar vector fields of the most degenerated case which is the double Lotka-Volterra case. The perturbative analysis is made with the method of successive derivatives of return mappings. As usual, this involves relative cohomology of the first integral which is in that case a rational function. In this case, we have to deal with a kind of "relative logarithmic cohomology" already known in singularity theory. We succeed to compute the first bifurcation function by residue techniques around each centers and they differ from one center to the other.

In this paper, we consider the system x˙=y(1+x)2−ϵP(x,y), y˙=−x(1+x)2+ϵQ(x,y) where P(x,y) and Q(x,y) are arbitrary quadratic polynomials. We study the maximum number of limit cycles bifurcating from the periodic orbits by using the Melnikov function of any order. We prove that the upper bound for the number of limit cycles is 3 and reached.