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## Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential Equations

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Research Article | Open Access

Volume 2013 |Article ID 575390 | https://doi.org/10.1155/2013/575390

Yanqin Xiong, Maoan Han, "Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System", Abstract and Applied Analysis, vol. 2013, Article ID 575390, 19 pages, 2013. https://doi.org/10.1155/2013/575390

# Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

Accepted21 Nov 2012
Published28 Feb 2013

#### Abstract

This paper concerns limit cycle bifurcations by perturbing a piecewise linear Hamiltonian system. We first obtain all phase portraits of the unperturbed system having at least one family of periodic orbits. By using the first-order Melnikov function of the piecewise near-Hamiltonian system, we investigate the maximal number of limit cycles that bifurcate from a global center up to first order of .

#### 1. Introduction and Main Results

Recently, piecewise smooth dynamical systems have been well concerned, especially in the scientific problems and engineering applications. For example, see the works of Filippov [1], Kunze [2], di Bernardo et al. [3], and the references therein. Because of the variety of the nonsmoothness, there can appear many complicated phenomena in piecewise smooth dynamical systems such as stability (see [4, 5]), chaos (see [6]), and limit cycle bifurcation (see [7â€“10]). Here, we are more concerned with bifurcation of limit cycles in a perturbed piecewise linear Hamiltonian system: where is a sufficiently small real parameter, with , , , and real numbers satisfying , and with compact. Then system (1) has two subsystems which are called the right subsystem and the left subsystem, respectively. For , systems (5a) and (5b) are Hamiltonian with the Hamiltonian functions, respectively,

Note that the phase portrait of the linear system with has possibly the following four different phase portraits on the plane (see Figure 1).

Then, one can find that system (1) can have 13 different phase portraits (see Figure 2) when at least one family of periodic orbits appears.

We remark that in Figure 2,â€‰GC: global center,â€ƒâ€‰Ho: homoclinic, â€ƒâ€‰He: heteroclinic,â€‰: center in the region , â€ƒâ€‰: center in the region ,â€‰: saddle in the region ,â€‰: saddle in the region ,â€‰: curvilinear or straightline in the region ,â€‰: curvilinear or straightline in the region .

It is easy to obtain the following Table 1 which shows conditions for each possible phase portrait appearing above. Also, cases (3), (5), (7), (9), and (13) in Figure 2 are equivalent to cases (2), (6), (8), (10), and (12), respectively, by making the transformation together with time rescaling .

 Coefficient conditions (1) (2) â€‰ (5) (3) (4) (7) (9) â€‰ (8) â€‰ â€‰ â€‰ â€‰ â€‰ â€‰ (11) ( ), (6) (10) â€‰ (12) ( ), â€‰ â€‰ â€‰ â€‰ (13) ( )

The authors Liu and Han [7] studied system (1) in a subcase of the case (1) of Figure 2 by taking . By using the first order Melnikov function, they proved that the maximal number of limit cycles on PoincarÃ© bifurcations is up to first-order in . The authors Liang et al. [8] considered system (1) in the case (5) of Figure 2 by taking , ,, and . By using the same method, they gave lower bounds of the maximal number of limit cycles in Hopf, and Homoclinic bifurcations, and derived an upper bound of the maximal number of limit cycles bifurcating from the periodic annulus between the center and the Homoclinic loop up to the first-order in . Clearly, the maximal number of limit cycles in the case (7) or (8) of Figure 2 is on PoincarÃ©, Hopf and Homoclinic bifurcations up to first-order in , by using the first order Melnikov function.

This paper focuses on studying the limit cycle bifurcations of system (1) in the case (1) of Figure 2 by using the first order Melnikov function. That is, system (1) satisfies Clearly, system (1) satisfying (9) has a family of periodic orbits such that the limit of as is the origin. The intersection points of the closed curve with the positive -axis and the negative -axis are denoted by and , respectively. Let Then, from Liu and Han [7], the first-order Melnikove function corresponding to system (1) is Let denote the maximal number of zeros of for and the cyclicity of system (1) at the origin. Then, we can obtain the following.

Theorem 1. Let (9) be satisfied. For any given , one has Table 2.

 â€‰ , , , , , , , , , , , , â€‰ â€‰ â€‰ ,

This paper is organized as follows. In Section 2, we will provide some preliminary lemmas, which will be used to prove the main results. In Section 3, we present the proof of Theorem 1.

#### 2. Preliminary Lemmas

In this section, we will derive expressions of in (11). First, we have the following.

Lemma 2. Suppose system (1) satisfies (9). Then,â€‰â€‰ in (11) can be written as â€‰â€‰where â€‰with â€‰â€‰â€‰ in (11) can be expressed as â€‰where â€‰with

Proof. We only prove (i) since (ii) can be verified in a similar way. By (11), we obtain which follows that by Green formula and (3) where Then, by Green formula again where By (3), (4), and the above formulas, we have Combining (20)â€“(25) gives (13) and (14). Thus, the proof is ended.

Then, using Lemma 2 and (6) we can obtain the following three lemmas.

Lemma 3. If , then in (11) has form where â€‰If , then we have where

Proof. Note that along . Then, inserting it into (14) follows that Let . Then we have and the above integral can be carried into Thus, using (30) and the above equation we can write (13) as where which gives (i) by (15). Thus, (i) holds and we can prove (ii) in the same way by (16)â€“(18). This ends the proof.

Lemma 4. Let system (5a) satisfy (3) and (4). Then â€‰ If has the expression â€‰where â€‰ If can be written as â€‰or â€‰where â€‰each is a nonzero constant and are polynomials of degree , respectively.

Proof. Since along the curve in (14) becomes where Let . Then, we have and the above equation becomes
For , make the transformation . Then, we have by (41) Substituting the above formula into (37), together with (13), gives that where Thus, by (15) and the above discussion we know that (i) holds.
For , (41) can be represented as where Recall that Then, by (46) and the above equation we obtain that It follows that where which is a polynomial of degree in . For convenience, introduce Then, combining (49) and (51) gives that Further, by using the formula we have that It follows that where which is a polynomial of degree in . Let Then, we have that by (55) and the above Substituting the above equation into (52), one can find that where for odd, for even, and which is a polynomial of degree in . Combining (37), (45), and (59) gives that which implies (35), together with (13) and (15).
Note that Then, we have Inserting the above formula into (35), we can obtain (36). Hence, the proof is finished.

Similar to Lemma 4, we can obtain the following lemma about .

Lemma 5. Let system (5b) satisfy (3) and (4). Then
If , in (11) has the expression where If in (11) has the form or where each is nonzero constant and are polynomials of degree , respectively.

#### 3. Proof of Theorem 1

In this section, we will prove the main results. Obviously, under (9) there are the following 9 subcases:(1), (2),(3), (4), (5), (6), (7), (8), (9).

We only give the proof of Subcases 1, 2, 3, 4, 5, and 6. And the Subcases 7, 8, and 9 can be verified, similar to Subcases 3, 4, and 5, respectively.

Subcase 1. . From (12) and Lemma 3, one can obtain that