On the Number of Limit Cycles of a Piecewise Quadratic Near-Hamiltonian System

and Applied Analysis 3


Introduction and Main Results
Recently, piecewise smooth systems attracted many researchers' attention since many real processes and different modern devices can be modeled by them; see [1,2] and references therein.Due to their nonsmoothness, these systems can have richer dynamical phenomena than the smooth ones (see [3][4][5] and the references cited therein).For instance, nonsmooth system can have the sliding phenomena and in [6,7] Giannakopoulos and Pliete have studied the existence of sliding cycles and sliding homoclinic cycles for a planar relay control feedback systems.
As we have seen, studying the existence and number of limit cycles is one of the main problems for piecewise smooth systems; see [8][9][10][11][12][13][14][15][16].Limit cycles of piecewise smooth linear differential systems defined on two half-planes separated by a straight line  = 0 or  = 0 have been studied recently in [8][9][10][11], from which one can find that 3 limit cycles can appear for piecewise smooth linear systems.From [12][13][14], we can know that piecewise linear and quadratic near-Hamiltonian systems can have 2 and 3 limit cycles, respectively.In fact, these papers investigated the problem for limit cycle bifurcations of piecewise linear Hamiltonian systems under piecewise polynomial perturbations of degree , obtaining some new and interesting results.Furthermore, the authors of [16] studied a piecewise quadratic Hamiltonian system (one side linear and another side quadratic) perturbed inside the class of piecewise polynomial differential systems of degree  and achieved that it can have 2 + [( + 1)/2] limit cycles, which implies that piecewise quadratic near-Hamiltonian systems can have 5 limit cycles.The authors of [15] also obtained this result by perturbed quadratic polynomial differential systems containing an isochronous center with piecewise quadratic polynomials of degree 2.
In this paper, we mainly study the problem for the maximal number of a quadratic near-Hamiltonian system.That is to say, we consider a system of the form where  > 0 is a small parameter, with  2  2 ̸ = 0, and Abstract and Applied Analysis Theorem 1.There exists a system of form (1) which has 8 limit cycles.
In the next section, the proof of Theorem 1 is presented.

Proof of Theorem 1
In this section, we divide the proof into two subsections: one section is to present some preliminary knowledge, which is useful to verify our main result; another section is to give the process of the proof of Theorem 1.
Lemma 2. The function  − 3 (ℎ) in (12) has the following expression: where Proof.Denote by  a disk of diameter  0 > 0 with its center at  1 .Then, for  0 > 0 small, rewrite  − 3 (ℎ) =  1 (ℎ) +  2 (ℎ), where with Note that  2 (ℎ) is a  ∞ function for ℎ 1 − ℎ > 0 small.Thus, it suffices to investigate the expansion of  1 (ℎ) for 0 < ℎ 1 − ℎ ≪ 1.To do this, we make a variable transformation Then system (4a) becomes where For  = 0, the Hamiltonian function of (18) has the form and the corresponding first order Melinikov function can be denoted by M(ℎ).By Remark 3.1.4of [18], we can know that  − 3 (ℎ) = M(ℎ), and then, from Corollary 3.2.2 and the Theorem 3.2.1 of [18],  1 can be written as where Thus, by the above discussion, one can obtain the expression of ( 13) and the formulas for  1 and  3 .Now, we give the expressions of  0 and  2 in (13).
In the same way, one can obtain the expressions of  + 3 (ℎ).
Then, using Lemmas 4 and 5, we can have the following.Theorem 6.Let (H) and (7) hold.Then if there exists  0 such that then system (1) can have 7 limit cycles near .
Proof.By the assumption and from the expansions in Lemmas 4 and 5 we have Note that  01 ,  02 ,  0 + c0 ,  1 + c1 , and  2 + c2 can be taken as free parameters.Then we can vary them one by one to change the signs of  1 ,  2 ,  3 , and .
In a word, we have proved that if then  1 ,  2 ,  3 , and  have one zero, one zero, two zeros, and three zeros which implies that seven limit cycles can appear in the neighbourhood of .

Proofs.
In this section, we will use Theorem 6 to prove our main result.For that purpose, let (46) Under (44), it is not hard to obtain the expressions for  1 , c1 ,  3 , and c3 in Lemmas 2 and 3.