Abstract and Applied Analysis
Volume 2014 (2014), Article ID 385103, 10 pages
http://dx.doi.org/10.1155/2014/385103

Research Article

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

Jing Tian

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 24 June 2014; Accepted 23 July 2014; Published 14 October 2014

Academic Editor: Tonghua Zhang

Copyright © 2014 Jing Tian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with the problem for the maximal number of limit cycles for a quadratic piecewise near-Hamiltonian system. By using the method of the first order Melnikov function, we find that it can have 8 limit cycles.

1. 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 [3–5] 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 [8–16]. Limit cycles of piecewise smooth linear differential systems defined on two half-planes separated by a straight line or have been studied recently in [8–11], from which one can find that 3 limit cycles can appear for piecewise smooth linear systems. From [12–14], 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 limit cycles, which implies that piecewise quadratic near-Hamiltonian systems can have 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 is a small parameter,

with , and

Then, our main results can be stated as follows.

Theorem 1. There exists a system of form (1) which has limit cycles.

In the next section, the proof of Theorem 1 is presented.

2. 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.

2.1. Preliminaries

Clearly, system (1) has the following two subsystems:

which are called left and right subsystems, respectively. And systems (4a) and (4b) are Hamiltonian with Hamiltonian functions, respectively,

For systems (4a) and (4b), we make the following assumption:

(H),  .

Then, under the condition (H), for , system (4a) (resp., (4b)) has a hyperbolic saddle (resp., ) and an elementary center (resp., ), where

We will assume that

since, for , we can make a variable transformation , , together with a time rescaling such that the resulting system satisfies (7). Further, note that there exist 2 possible cases for level curves of on the plane, where , ; see Figure 1.

(a)

(b)

Figure 1: Level curves of .

We remark that, in Figure 1, , . Then, one can obtain 9 possible phase portraits of system (1) under (H) and (7) when it has a generalized closed orbit and two hyperbolic saddles; see Figure 2. For system (1), a generalized closed orbit passes through the -axis two times.

(a) ,

(b) ,

(c) ,

(d) ,  ,  

(e) ,  ,  

(f) ,  ,  

(g) ,

(h) ,

(i) ,

Figure 2: Possible portraits of (1) with two saddles and a generalized closed orbit.

Now, we study limit cycle bifurcations on system (1). For simplicity, we consider the case

that is,

Then, the phase portrait of system (1) is Figure 2(h), and there are four families of periodic orbits given by

where , and , . Further, we have

see Figure 3. From (11), one can know that, as approaches , the limit of is denoted by , which is a generalized polycycle with two hyperbolic saddles.

Figure 3: Phase portrait of system (1) under (9).

Then, from Theorem 1.1 of [17], for system (1), we have four first order Melnikov functions as follows:

where , , , , , and are given in (10).

Now, we deduce the expansion of ,  , and near in the above. First, we have the following.

Lemma 2. The function in (12) has the following expression:

where

Proof. Denote by a disk of diameter with its center at . Then, for small, rewrite , where

with

Note that is a function for small. Thus, it suffices to investigate the expansion of for . To do this, we make a variable transformation

Then system (4a) becomes

where

For , the Hamiltonian function of (18) has the form

and the corresponding first order Melinikov function can be denoted by . By Remark of [18], we can know that , and then, from Corollary and the Theorem of [18], can be written as

where

Thus, by the above discussion, one can obtain the expression of (13) and the formulas for and . Now, we give the expressions of and in (13).
On the curve intersect -axis with two points denoted by and . Then, from (12), we have

where

By Green’s formula two times to , we obtain

Further, one has

Thus, we have

Then, let . Then, by (11), we can obtain the expression of in (13). Further, differentiating (13) with respect to , together with Lemma of [18], we obtain

which, as , gives that

Let in the above. Then we can easily obtain the formula for in (13). This ends the proof.

In the same way, one can obtain the expressions of .

Lemma 3. For in (12), one has

where

By applying Theorem of [18], together with the proof of Lemma 2, we have the following.

Lemma 4. The functions , , given in (12) can be expressed as

where , , , are given in Lemmas 2 and 3, and

By above three Lemmas 2, 3, and 4, together with (12), one can derive the following.

Lemma 5. For and , one has

where the corresponding coefficients are given in (14), (32), and (34).

Then, using Lemmas 4 and 5, we can have the following.

Theorem 6. Let (H) and (7) hold. Then if there exists such that

then system (1) can have limit cycles near .

Proof. By the assumption and from the expansions in Lemmas 4 and 5 we have

Note that , , , , and can be taken as free parameters. Then we can vary them one by one to change the signs of , , , and .
First, let and . Then, in this case, we have

which means that there exists a zero with such that .
Second, fix and vary , so that

Hence, there exist ,   such that ,  .
Third, let ,  , implying

Then ,   for some ,  .
Fourth, let and ; we have

So there exists such that .
Finally, vary . In this case,

So there exists such that .
In a word, we have proved that if

then , , , and have one zero, one zero, two zeros, and three zeros which implies that seven limit cycles can appear in the neighbourhood of .

2.2. Proofs

In this section, we will use Theorem 6 to prove our main result. For that purpose, let

Then, the Hamiltonian functions in (5) can be rewritten as

And in this case

Under (44), it is not hard to obtain the expressions for , , , and in Lemmas 2 and 3.

Lemma 7. Under (44), for system (1), one has

For the other expressions of coefficients in Lemmas 4 and 5, one has the following.

Lemma 8. Under (44), for system (1), one has

Proof. Note that . Then, by Green’s formula two times, we derive

On the other hand, along the curve , we have , for . Then, (49) can be represented as

By applying Maple, one can compute that

Inserting the above into (50) gives the expression of .
Recall that if and only if . Thus, by (4a), we rewrite as

which, by Maple, yields the conclusion for . Using a similar way, one can derive the formulas for and . Now, we deduce the formulas for and .
Note that, on the curve ,  , we have

By Maple again, we obtain that

Thus, we can obtain the conclusion for .
When , we have . Thus, in (14) can be rewritten as

which, by Maple, implies the conclusion. Further, we can prove and in the same way to and , respectively. This finishes the proof.

Proof of Theorem 1. Let

Then, solve the equations

which, together with (56), gives that

where . Then, under (58)

Further,

Thus, by Theorem 6, there exist limit cycles near for small.
Under (58), we have for

Now, we derive the expansion of . To achieve this, make the transformation

together with time scaling . Then the system becomes

where

For , the Hamiltonian function of (63) has the form

Denote by the first Melnikov function of system (63) near , and note that

Then we have . From Theorem of [18], we can obtain

where, under (58),

Similarly, under (58) we have

This implies that there exists another zero between and for . This ends the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to give thanks to Professor Maoan Han for his helpful discussions during the preparation of the paper. The project was supported by National Natural Science Foundation of China (11271261), FP7-PEOPLE-2012-IRSES-316338, and Shanghai Municipal Natural Science Foundation no. 12ZR1421600.

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