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Bifurcation of Limit Cycles from a Focus-Parabolic-Type Critical Point in Piecewise Smooth Cubic Systems

“…By using the generalized polar coordinates to compute the Lyapunov constants V n , we obtain nine weak focus conditions of order 9 and eleven weak center conditions at (0, 0). In addition, we also obtain five conditions under which (0, 0) may be a weak center by direct numerical simulation for system (2). Then, we prove that at least nine limit cycles can be bifurcated by applying linear perturbations.…”

“…) is called a weak focus of order k of system (2). Instead, we call (0, 0) a weak center of (2) if V j = 0 for all j ≥ 1.…”

“…In particular, the second part of Hilbert's 16th problem, which is one of the 23 problems proposed by D. Hilbert, has attracted much attention from many scholars. Motivated by these problems, many works, for example, [1][2][3], have been devoted to studying the number of limit cycles and finding a uniform upper bound of the limit cycles of planar systems of the degree of n. It is a pity that this problem is far from solved for n = 2. Over the past few decades, many mathematicians have been interested in the extension of the second part of Hilbert's 16th problem and paid attention to two related but weaker problems, namely the center-focus and cyclicity problems, and further found the maximum number of small amplitude limit cycles.…”

Bifurcation of limit cycles for a family of perturbed Kukles differential systems

“…Bifurcation of limit cycles in Kukles systems have been tackled by several authors and by using different approaches. See for example [1,6,12,13,15].…”

“…Moreover, the transformed system can be written in the form (9) by changing the independent variable t by the variable θ. Since r(θ; r 0 ), given in (12), is the periodic solution of the unperturbed system, the first order averaged function (13) becomes…”