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

Abstract

This paper contains two parts. In the first part, we shall study the Abelian integrals forŻo ladek's example [13], in which it is claimed the existence of 11 small-amplitude limit cycles around a singular point in a particular cubic vector filed. We will show that the basis chosen in the proof of [13] are not independent, which leads to failure in drawing the conclusion of the existence of 11 limit cycles in this example. In the second part, we present a good combination of Melnikov function method and focus value (or normal form) computation method to study bifurcation of limit cycles. An example by perturbing a quadratic Hamiltonian system with cubic polynomials is presented to demonstrate the advantages of both methods, and 10 smallamplitude limit cycles bifurcating from a center are obtained by using up to 5th-order Melnikov functions.

An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

“…Also note that besides the perturbation parameters δ involved in p (x, y, ε, δ) and q (x, y, ε, δ), there are also parameters µ included in the Hamiltonian function H (x, y, µ), which can also be used to increase the number of bifurcating limit cycles. In the following, we first show the equivalence of our method and the Melnikov function method [Tian & Yu, 2016, and then show why the free parameter involved in the Hamiltonian function can be used to get more limit cycles.…”

“…The system considered in [Żo ladek, 1995] was reinvestigated by Yu and Han [2011] using the method of focus value computation, and only nine small limit cycles were obtained. Recently, Tian and Yu [2016] found the mistakes in [Żo ladek, 1995] and showed that the example given in [Żo ladek, 1995] indeed only has nine limit cycles using up to second-order Melnikov functions. In a very recently published paper [Tian & Yu, 2018], the authors applied high-order analysis to prove that the example given byŻo ladek [1995] indeed can have 11 small-amplitude limit cycles if at least seventh-order analysis (equivalent to seventh-order Melnikov function method) is used.…”

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

“…This obvious difference motivated a further investigation on this problem. Very recently, Tian and Yu [24] has proved that the 11 limit cycles obtained by _ Zoła¸dek [20] are not correct, and the mistakes leading to the erroneous result have been identified.…”

Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis

“…Later, Yu and Han applied the normal form computation method and got only 9 limit cycles around E 0 [24] by analyzing the ε-and ε 2 -order focus values. Recently, it has been shown [25] that errors are made in [23] for choosing 12 integrals as the basis of the linear space of corresponding Melnikov functions of system (11)| ε=0 . In fact, among the 12 chosen integrals, two of them can be expressed as linear combinations of the other ten integrals, and therefore only 9 limit cycles can exist, agreeing with that shown in [24].…”