Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory

In this paper we classify the phase portraits in the Poincar\'e disc of the centers of the generalized class of Kukles systems $$=-y,=x ax^3y bxy^3,$$ symmetric with respect to the y-axis, and we study, using the averaging theory up to sixth order, the limit cycles which bifurcate from the periodic solutions of these centers when we perturb them inside the class of all polynomial differential systems of degree 4

An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems: the Averaging Method Revisited

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows to transform the considered differential systems to the normal formal of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms that determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.Comment: Proc. 44th ISSAC, July 15--18, 2019, Beijing, Chin

Limit cycles bifurcating from a family of reversible quadratic centers via averaging theory

Consider the class of reversible quadratic systems x· = y, y· = -x + x²+ y² - r², with r > 0. These quadratic polynomial differential systems have a center at the point ((1 -√(1+4r²)/2, 0) and the circle x² + y² = r² is one of the periodic orbits surrounding this center. These systems can be written into the form x· = y + (4 + A)x² - Ay², y· = -x, with A ϵ (-2, 0). For all A ϵ R we prove that the averaging theory up to seventh order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles bifurcating from the periodic orbits surrounding the center (0,0) of that system. Up to now this result was only known for A = -2 (see [22, 23])

The local cyclicity problem : Melnikov method using Lyapunov constants

In 1991, Chicone and Jacobs showed the equivalence between the computation of the first-order Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, so M(6) ≥ 44. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that Mcp(4) ≥ 43 and Mcp(5) ≥ 65

Limit cycles of a perturbation of a polynomial Hamiltonian systems of degree 4 symmetric with respect to the origin

We study the number of limit cycles bifurcating from the origin of a Hamiltonian system of degree 4. We prove, using the averaging theory of order 7, that there are quartic polynomial systems close these Hamiltonian systems having 3 limit cycles