Bifurcation of Limit Cycles and Center Conditions for Two Families of Kukles-Like Systems with Nilpotent Singularities

We solve theoretically the center problem and the cyclicity of the Hopf bifurcation for two families of Kukles-like systems with their origins being nilpotent and monodromic isolated singular points. 1. Introduction One of the main open problems in the qualitative theory of planar analytic differential systems is the case when the singular point is monodromic (the orbits move around the singular point). A monodromic point of an analytic system is either a center (i.e., a singular point with a punctures neighborhood filled with periodic orbits) or a focus (i.e., a singular point with a neighborhood where all the orbits are spirals which arrive at the equilibrium point in forward or backward time). The problem of distinguishing when a monodromic singular point is either a center or a focus is called the center problem. The so-called center problem for planar vector fields has been intensively and extensively studied over the last century and is also closely related to Hilbert’s 16th problem. The singular point can be elementary or not in the sense whether the corresponding Jacobian matrix has zero eigenvalues. If the eigenvalues of the quoted matrix are imaginary with real part null, the singular point may be a focus or a center, which is known as the celebrated Poincaré-Lyapunov center problem and has been theoretically solved by Poincaré [1] and Liapunov [2]. If the matrix of the linear part at the singular point has its two eigenvalues equal to zero, but it is not identically null, by Andreev [3] we know what is the behavior of the solutions in a neighborhood of the singular point, except if it is a center of a focus (nilpotent center problem). Yet for the bifurcation of limit cycles and center problem of nilpotent singular points in a planar vector field, its intrinsic dynamics is still far away from understanding due to the complexity and technical difficulties. Therefore, it is natural to restrict our study to nilpotent singularities. An analytic system of differential equations in the plane having an isolated nilpotent singularity, in some suitable coordinates, can be written as with and real analytic functions without constant nor linear terms defined in a certain neighborhood of the singularity. The study of nilpotent singularities and their unfolding for vector fields is important not only for mathematical interest but also for practical reasons. To solve the finite cyclicity problem in the second part of Hilbert’s 16th problem, Dumortier et al. [4, 5] presented an impressive list of 121 graphics that occur in quadratic systems, among which many