Four Limit Cycles From Perturbing Quadratic Integrable Systems by Quadratic Polynomials

Abstract

In this paper, we give a positive answer to the open question: Can there exist 4 limit cycles in quadratic near-integrable polynomial systems? It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least 4 limit cycles with (3, 1) distribution. The method of Melnikov function is used.

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

“…h 3. 12 Limit cycles in system (6) Now we return to system (6) and let the parameter a be free to vary. In this case, we have the following result.…”

“…For general quadratic polynomial systems, the best result is Hð2Þ P 4, obtained more than 30 years ago [4,5]. Recently, this result was also obtained for near-integrable quadratic systems [6]. However, whether Hð2Þ ¼ 4, is still open.…”

“…In the next section, we use the method of focus value computation to show that there are 11 small-amplitude limit cycles around the origin of the systems (8) and (9). In Section 3, we use system (6) with the free coefficient a to prove that there exist 12 small-amplitude limit cycles around the origin. Conclusion is drawn in Section 4.…”

Bifurcation of ten small-amplitude limit cycles by perturbing a quadratic Hamiltonian system with cubic polynomials

“…Then, we may determine the number of the limit cycles of system (6) emerging from the closed ovals {Γ h } by studying the zeros of the first non-vanishing Melnikov function (8). Denote Z(n) the sharp upper bound of the number of zeros of M 1 (h) for system (6), where n = max{deg(P ), deg(Q)}. Gavrilov [18] proved Z(2) = 2 for the Hamiltonian H with four distinct critical values (in a complex domain).…”

Limit cycle bifurcations by perturbing a class of integrable systems with a polycycle

“…However, in recent years, there were some works on limit cycle bifurcations by perturbing non-Hamilton integrable systems (see [7,13,[15][16][17] for example). As we know, the main tool for studying the bifurcation problem of limit cycles is to use the first order Melnikov function.…”