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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 690453, 27 pages
Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
Received 26 April 2012; Accepted 8 August 2012
Academic Editor: Alberto D'Onofrio
Copyright © 2012 Yanqin Xiong and Maoan Han. 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.
We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin.
1. Introduction and Main Results
As well known, Liénard systems describe the dynamics of systems of one degree of freedom under existence of a linear restoring fore and a nonlinear dumping. In the first half of the last century models based on the Liénard system were important for the development of radio and vacuum tube technology. Nowadays the system is widely used to describe oscillatory processes arising in various studies of mathematical models of physical, biological, chemical, epidemiological, physiological, economical, and many other phenomena (see Glade et al. , Llibre  and references therein). Further, quadratic systems and some other systems can be transformed into Liénard systems by suitable changes, see for instance Han et al. , Cherkas , Gasull , Giné, and Llibre . As we have seen, the main concern on Liénard systems is the center and focus problem and the number of limit cycles, see [7–17], and references therein. Here, we briefly list some known results related to our study in this paper. Consider a Liénard system of the form where we suppose that the functions and satisfy two assumptions below.(I)There exists a positive number such that is continuous for with and is continuous for with , and existing.(II)For , , where and . Obviously, if , system (1.1) has a singular point at the origin. By Han and Zhang , the origin is a generalized singular point if . For example, the following system: has a generalized center at the origin.
Under the condition (I), the equation has two solutions and for . Let . Then in 1952, Filippov proved the following theorem (see Chapter 5 of Ye ).
In 1985, Han  obtained the following.
Theorem 1.2. Consider the system where , and are continuous functions satisfying(i) with ,(ii) for and where . Let satisfy for . Then system (1.4) has a stable focus (resp., center, unstable focus) if for .
For center conditions, in 1976, Cherkas  proved the following.
From Cherkas , one can see that the above result is also true for analytic system (1.1). In 1998, Gasull and Torregrosa  studied the center problem for analytic systems of form (1.4) using the Cherkas’ method, generalized Theorem 1.3 to (1.4) and presented interesting applications to some polynomial systems. Then in 2006, Cherkas and Romanovski  gave a necessary and sufficient condition for a Liénard system with nonlinearities of degree six to have a center at the origin. About Theorem 1.2, we have the following two remarks.
Remark 1.5. If and are with . Then in (1.6) has the form In this case, Han  proved that if for , then , and the origin is a focus of order if in addition. Further, if a vector parameter appears in , then and for all and any .
In this paper, we consider the following discontinuous Liénard system: where with being an integer, is a function satisfying and where and are on , and , respectively, with . Further, suppose there exist integers , , , such that for and for .
Let where for and , respectively.
This paper is devoted to studying the local property of system (1.13) at the origin and the number of limit cycles bifurcated from the origin as varies. The authors  considered system (1.13) for the case and studied the center focus problem. It is easy to prove that for system (1.13) under (1.15) and (1.16), the condition (ii) of Theorem 1.2 is equivalent to the following: where . For convenience, introduce where are relatively prime numbers, that is, . Then it is easy to see that there must exist an integer such that is an integer, and our main results are the following.
Theorem 1.6. Let (1.15), (1.16), (H1), and (H2) hold. Then(i) can be expressed as where satisfies for small and for with .(ii) If there exist and such that and where , then system (1.13) has cyclicity near the origin for all near .
In many cases, the function in (1.13) is linear in . Then the coefficients in (1.21) are linear in . Let there exist and integer such that which implies that the linear equations , of have a unique solution of the form for near . Obviously, is linear in . Further, let for some integer . Then, we can obtain the following.
Theorem 1.8. Consider system (1.13), where the function is linear in . Suppose (1.15), (1.16), (H1), and (H2) hold. Let there exist integers and points and such that (1.24) and (1.26) hold with (i) If and then system (1.13) has cyclicity limit cycles near the origin for all near .(ii) If and as , , with then system (1.13) has cyclicity limit cycles at the origin for all near .
The conclusion (i) of Theorem 1.8 can be proved in a similar manner to the proof of Theorem 2 of  and (ii) can be verified by Theorem 1.7. Thus, we do not verify it here. The proof of Theorems 1.6 and 1.7 is presented in Section 2. In Section 3, we give some applications.
In this section, we verify Theorems 1.6 and 1.7. Before proving them, we need to introduce a bifurcation function of system (1.13) and establish some preliminary lemmas, which will be used in our proof. First, we have the following lemma.
Proof. Consider the equation for and . By (1.18), we can obtain where . The implicit function theorem implies that the equation has a unique solution for small with . Let . Then, for sufficiently small is the solution of (2.2). Combining (2.3) and (2.4), the above formula can be represented as with . Thus, the proof is ended.
Following the idea of Han , we have the following lemma.
Proof. Let and make the transformation together with the scaling of the time to system (1.13) so that (1.13) can be changed into where , and denote the inverse of the transformation (2.8). In fact, by (1.18) and similar to the proof of Lemma 2.1, we can obtain For , by (1.15) and (2.10), we have where . Similarly, for , by (1.16) and (2.10), we can obtain where . This ends the proof.
Proof. By (1.15), (1.16), and Lemma 2.1, we can have
Since is an integer, the above formula can be written as
where . Then we express (2.16) with respect to the power of in ascending order, yielding the form of (1.21). In addition, for defined in Theorem 1.6 there must exist some integers and such that . Hence, we can obtain
This completes the proof of the conclusion (i).
Now we prove the conclusion (ii). For sufficiently small, (2.8) and (2.10) imply that Then noting that , we can get By (2.11), (2.12) and the above formula, we have for sufficiently small Let Combining (2.10) and (2.21) gives that Substituting the above formula into (1.21) or (2.16) yields that Note that where with and . Similarly, where Combining (2.23), (2.24), and (2.26), we can obtain Similar to (1.21) and (2.16), the above formula can be rewritten as where from (2.17) From (2.25), (2.27), and the above equation, in (2.29) can be rewritten as the form of (2.14). Further, combining (2.17), (2.20), and (2.29) yields (2.13). Thus, the proof is ended.
Next, we establish the bifurcation function of system (1.13). For the purpose, let . Then (2.6) can be translated into In a similar way to the proof of Lemma 2.3 of , it is easy to prove that for small under (H1) and (H2), which shows that in (2.31) is well defined. Introduce Then by (2.31) and (2.33), we can obtain where . Clearly, (2.34) is a -periodic equation in . By (2.7), we have From (2.33) and (2.35), it is easy to see that is in for any given . Let denote the solution of (2.34) satisfying . For convenience, let Define a function as follows: It is obvious that system (1.13), (2.6), (2.31), or (2.34) has a periodic orbit near the origin if and only if in (2.37) has a zero in for sufficiently small. See Figure 1.
Proof. Denote by . Then and it satisfies Let . Then we can obtain that is a solution of the equation satisfying . Let . Then and where Thus, by (2.43), (2.44) we have easily by the mean value theorem, where . Further by (2.34) and (2.42), we can obtain which follows that by Lemma 2.3 where is a function in with which implies by the above discussion. Then it follows from (2.13), (2.45), and (2.48) that where are functions in with which shows that for all . Since and satisfy (2.42) and (2.34), respectively, using (2.46) we have where and . Then noting (2.40), applying the formula of variation of constants to the above equation yields Inserting (2.50) into the above formula and taking give by (2.37) that which implies (2.38) by (2.33). Hence, The proof is completed.
Proof of Theorem 1.6. One can see that the conclusion (i) is true by Lemmas 2.1 and 2.3. Now, we prove the conclusion (ii). Obviously, it suffices to prove the following two points.(1)There are at most limit cycles near the origin for all near .(2)There can appear limit cycles in any given neighborhood of the origin for some arbitrarily close to .
In fact, noting that By our assumption, we have from (2.38) and (2.55) where , which implies in (2.56) are in satisfying for sufficiently close to .
We claim that in (2.56) has at most zeros in small for . We can proceed with the proof by induction on .
First, when , from (2.56), we can obtain Note that which implies for and near since by (2.55). Hence, by Rolle’s theorem, the conclusion (1) is true for .
Assume that the conclusion (1) is true for . That is by (2.56) has at most zeros for and sufficiently small with . Let us prove that the conclusion (1) is also true for . Then from (2.56), we can obtain where , , . Then, where , , satisfying and is in with , which implies that has at most zeros in small for near by induction assumption since . Hence, by Rolle’s theorem, it follows that in (2.62) has at most zeros in for near . This ends the proof of conclusion (1).
Now, we verify the conclusion (2). For simplicity, we can assume . Further, by (1.22), without loss of generality, suppose Then we can fix for since . Consider the equations By the implicit function theorem, the equations have a unique set of solutions for near , which means can be taken as free parameters. Hence, by changing the sign of in turn such that which follows that in (2.56) satisfy This ensures that the bifurcation function has zeros in for some near , which implies the conclusion (2). This ends the proof.
Proof of Theorem 1.7. From the proof of Theorem 1.6, it is easy to see can be taken as free parameters by (1.22), which implies can also be taken as free parameters by the definition of in Lemma 2.4. From (1.23) and Lemma 2.4, we have for sufficiently small since if and only if , which means that the coefficients in (2.38) satisfy Since can be taken as free parameters, we can write where . Inserting (2.71) into (2.38) gives that where We can easily prove in (2.72) has at most zeros for near by induction in a similar way to the above proof. Further, we can choose satisfying which ensures that in (2.72) has zeros for near . The proof is finished.
Corollary 2.5. Let (1.15), (1.16), (H1), and (H2) hold. If there exist a point and an integer such that then the origin is a stable (resp., unstable) focus of system (1.13) for and there has at most limit cycles at the origin for all near .
Corollary 2.6. Let (1.15), (1.16), (H1), and (H2) hold. If there exists such that for , then for any given there exists a neighborhood of the origin such that for all , system (1.13) has at most limits cycles in .
Further, about system (1.13), we have the below two remarks in order.
Remark 2.7. If the function in (1.13) has the form with , we also can obtain the same conclusions since this system is equivalent to the following:
Example 3.1. First consider system (3.1) with where and , . By Lemma 2.1, it is clear that for small. For any given , denote the cyclicity of system (3.1) near the origin for . We study the cyclicity of system (3.1) under (3.2) near the origin with the following three cases.
Case 1. , . In this case, we claim that . In fact, by (3.2) and (3.3), we have for . From (3.4) it is easy to see that for if and only if for , which shows that by Corollary 2.6. Thus we only need to prove that . For , by (3.4), we can obtain