Limit Cycles Coming from Some Uniform Isochronous Centers

Abstract

Abstract. This article is about the weak 16-th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centerṡof degree 2n+3 and we perturb them inside the class of all polynomial differential systems of degree 2n + 3. For n = 0, 1 we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n = 2 we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center.

Algorithmic Averaging for Studying Periodic Orbits of Planar Differential Systems

“…Thus system (C.2) can have up to 3 limit cycles by using the first order averaging method from Theorem 2.3. Therefore, using our algorithmic approach we verified the first result in ( [34], Thm. 1.1).…”

“…In order to derive the interval D in this case, we will construct an equivalent solution set SIs of SIs that contains only the rational polynomial inequalities, and then use the SemiAlgebraic command in Maple to compute the solutions. Below we provide a concrete example to show the feasibility this algorithm, one may check the results in [34]. More experiments can be found in Section 4.…”

On extended Chebyshev systems with positive accuracy

“…Moreover we can also prove that 9 ≤ Z(F). Another direct application of this work can be found in [7] where the set of functions is an ET-system with accuracy 1. Finally in Section 5, as a nontrival application of the above results, we improve the results of [8] where the maximum number of limit cycles for a class of nonsmooth systems is studied.…”

On the number of limit cycles for a class of discontinuous quadratic differential systems

“…It is well known that one of the important open problems in the qualitative theory of real planar differential systems is the study of limit cycles. For about one century, scholars focus on the bifurcation of limit cycles in the continuous planar polynomial differential systems, see [1,2,3,4,5,6,7,12,13,14] and the references therein. Nevertheless, it is still open even for the quadratic cases.…”