Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 385419, 9 pages

http://dx.doi.org/10.1155/2013/385419

Research Article

Bifurcation of Limit Cycles of a Class of Piecewise Linear Differential Systems in with Three Zones

Yanyan Cheng

Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Received 1 March 2013; Accepted 16 April 2013

Academic Editor: Qingdu Li

Copyright © 2013 Yanyan Cheng. 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.

We study the bifurcation of limit cycles from periodic orbits of a four-dimensional system when the perturbation is piecewise linear with two switching boundaries. Our main result shows that when the parameter is sufficiently small at most, six limit cycles can bifurcate from periodic orbits in a class of asymmetric piecewise linear perturbed systems, and, at most, three limit cycles can bifurcate from periodic orbits in another class of asymmetric piecewise linear perturbed systems. Moreover, there are perturbed systems having six limit cycles. The main technique is the averaging method.

1. Introduction and Statement of the Main Result

Piecewise linear systems are used extensively to model many physical phenomena, such as switching circuits in power electronics [1, 2] and impact and dry frictions in mechanical systems [3]. These systems exhibit not only standard bifurcations but also complicated dynamical phenomena not existing in smooth systems. The study and classification of various kinds of bifurcation phenomena for piecewise linear systems have attracted great attentions since the last century, see, for example, [4, 5] and the references therein.

In recent years, many papers studied the bifurcation of limit cycles and the number and distribution of these limit cycles. Most of them studied the planar piecewise linear system, see for example, [6–9] and the references quoted there. There are also some papers which studied bifurcation of limit cycles of 3D piecewise linear systems [10, 11]. For high-dimensional cases, there are a few papers [12–16]. Especially in [12] the authors studied the bifurcation of limit cycles of a class of piecewise linear systems in . They showed that three is an upper bound for the number of limit cycles that bifurcate from periodic orbits.

In this paper, we study the limit cycles bifurcated from periodic orbits of a linear differential system in    when the perturbation is piecewise linear with two switching boundaries. We consider two classes of asymmetric perturbation. With the first class of asymmetric perturbation, six is the upper bound for the number of limit cycles bifurcated from periodic orbits, and there are perturbed systems having six limit cycles. With the second class of asymmetric perturbation, three is the upper bound for the number of limit cycles bifurcated from periodic orbits, which generalizes the result of the paper [12].

More precisely, we study the maximum number of limit cycles of the 4-dimensional continuous piecewise linear vector fields with three zones of the form

for sufficiently small real parameter, where

and   is given by

with  ,  ,  and    the piecewise linear function

 if  ,

 if  ,

 if  ,

where  . The independent variable is denoted by  ; vectors of    are column vectors, and    denotes a transposed vector.

For  ,  system (1) becomes

Our main results are the following.

Theorem 1. If  , six is the upper bound for the number of limit cycles of system (1) which bifurcate from the periodic orbits of system (7) with    sufficiently small. Moreover, there are systems of form (1) having six limit cycles.

Theorem 2. If  , three is the upper bound for the number of limit cycles of system (1) which bifurcate from the periodic orbits of system (7) with    sufficiently small. Moreover, there are systems of form (1) having three limit cycles.

It is worth to note that Theorem 2 generalizes the result of paper [12]. The method for computing the number of limit cycles bifurcated from periodic orbits is the averaging method, which is obtained by Buică and Llibre [17]. By means of the result of paper [18], we can study the stability of the limit cycles of Theorem 1; for more details see Remark 10.

Theorems 1 and 2 will be proved in Section 3. In Section 2, we review the results from the averaging theory necessary for proving these two theorems. Further discussions on the number of limit cycles of the perturbed system are present in Section 4. There is a conclusion given in the last section.

2. First-Order Averaging Method

The aim of this section is to review the first-order averaging method which is obtained by Buică and Llibre [17]. The advantage of this method is that the smoothness assumptions for the vector field of the differential system are minimal.

Theorem 3 (see [17]). Consider the following differential system:

where  ,    are continuous functions. T-periodic in the first variable, and    is an open subset of  . We define    as

and assume that

(i) and    are locally Lipschitz with respect to  ;(ii)for    with  , there exists a neighborhood    of    such that    for all    and  .

Then, for    sufficiently small, there exists an isolated T-periodic solution    of system (8) such that    as .

We remind here that    denotes the Brouwer degree of the function    with respect to the set    and the point  , as is defined in [19]. The following fact is useful for the proof of Theorems 1 and 2.

Fact 1. Let be a function, with , where is an open subset of and . Whenever is a simple zero of   (i.e., ), there exists a neighborhood of such that for all . Then, .

3. Proof of Main Theorems

The proof of Theorems 1 and 2 is based on the first-order averaging method presented in the previous section. In order to apply this method, we will first reduce the four parameters of the vector    in the definition of the function    to one, and then we will change the variables in order to transform the system into the standard form for the averaging method. After that, we will calculate the number of its isolated zeros.

Lemma 4. By a linear change of variables, system (1) can be transformed into the system

where    is an arbitrary matrix and    or  .

Proof. A linear change of variables  , with    invertible, transforms system (1) into

where  ,  .

We have to find    invertible which satisfies

It is easy to obtain that    has the following form:

Thus, we have

where

If  , it is easy to find    invertible with  ,  ,  ,    satisfying

If  , it is easy to find    invertible with  ,  ,  ,    satisfying

Changing  variables    to    with  , then we obtain system (10).

The standard form of the averaging method is obtained by changing variables to with

Thus, system (10) is transformed into the following system:

where  ,  ,  and    are given by

and for every  ,

where    are elements of the matrix    of Lemma 4.

We take    sufficiently small,    arbitrarily large and

Then, the vector of system (19) is well defined and continuous on  . Moreover, the system is  -periodic with respect to variable    and locally Lipschitz with respect to variables  . Our next step is to find the corresponding function  ,  , where

for  .

In order to calculate the exact expression of  , we denote

for  each  , where    is the piecewise linear function given by (4)–(6). Without loss of generality, we assume that the slope    of    is positive.

Lemma 5. The integrals    and    given by (24)-(25), respectively, have the following expressions:

and

(1) if  ,

(2) if  ,

(3) if    and  ,

(4) if    and  ,

(5) if    and  ,

 where

 for , and

The proof of this lemma is given in the appendix.

Remark 6. If    and  , system (1) can be transformed into the system which is studied in the paper [12].

Lemma 7. If  , one defines  ,    and consider the equation  ,   with    given by (24), and   is  a real parameter. Then,

(1)if    or  , the equation has no solutions;(2)if  , then the interval    is a continuum of solutions;(3)if , there is an unique solution  ;(4)if  , there are two solutions    and  .

Proof. If  , we have    and  . It is easy to see that all    are a solution if  . If    changing the variable    and defining  , we obtain the equivalent equation

with simple computation; we find that the function    is strictly monotonically increasing of variable    when    and strictly monotonically decreasing when  . The function    gets to the maximal value    when  . Also we have    as    and  . The proof is similar if  .

Lemma 8. If  , one defines    and  consider the equation  ,    with    given by (24), and   is  a real parameter. Then,

(1)if    or  , the equation has no solutions;(2)if  , then the interval    is a continuum of solutions;(3)if  , there is an unique solution  .

Proof. We only consider the case when  : the proof is similar when    and  . It is easy to see that all    are a solution if  . If    changing the variable    and defining  , we obtain the equivalent equation

With simple computation, we find that the function    is strictly monotonically increasing of variable  . It is easy to know    as    and    as  .

With Lemma 5, we obtain the expressions for the components of function  ,

where    are constants that depend linearly on  

According to Theorem 3 and Fact 1, for each simple zero    of (36) there is an isolated  -periodic solution      of system (19) with    sufficiently small such that    as . Any isolated  -periodic solution of system (19) with    sufficiently small corresponds to a limit cycle of system (10). Thus, the most important task is to calculate the number of the simple zeros of function  . We solve the two first equations of (36), then, we get

where

Substituting (38) into the third equation, we obtain

where

It is necessary to study the zeros of    instead of the zeros of  .

Lemma 9. The function    given by formula (41) can have at most six isolated zeros, and they appear in pairs  .

Proof. Substituting    and    in    we get

where

When we consider the case    and  ,    becomes

It follows that we have to find solutions of (42) or (44) in the interval  . This is equivalent to

which is the polynomial equation

This equation can have at most six roots in the interval  . Then,    has at most six solutions  .   Since    for all  , it is clear that if    is a zero of    then    is also a zero.

The functions  ,  ,  ,  and    have the properties  ,  ,  ,  and  . So, we have

Thus, the equation   at most three zeros that satisfy  . With Lemma 7 for a fixed  , we at most find two isolated value of    from  . With Lemma 8 for a fixed  , we at most find one isolated value of    from  . For fixed    and fixed  ,    gives at most one isolated value for  . Thus, we conclude that if    the maximum number of limit cycles for system (1) is six, and if    the maximum number of limit cycles for system (1) is three.

Remark 10. Using the main result of [18], the stability of the limit cycles associated with the solution    is given by the eigenvalues of the matrix

In order to show that there exist examples with exactly six limit cycles, we consider the following values of the coefficients:

More precisely, the system has the following form:

where

and    satisfy

It is easy to know  . Computing the six solutions of  , we get  . The values of  ,    and    are given in Table 1.

Table 1: The values of , , , and .

There are three values of    that satisfy    and  . These three solutions are  .

The six values of solution  ,  ,    and the value of the Jacobian at the solution    are given in Table 2.

Table 2: The values of solution and the Jacobian .

1.484
0.742
11.957

4.139
2.07

1.484
1.285

4.139
3.585

1.254
1.254
5.732

8.672
8.672

4. Conclusion

In this paper, we have studied the limit cycles bifurcated from periodic orbits of a linear differential system in    when the perturbation is piecewise linear with two switching boundaries. We considered two classes of asymmetric perturbation. We have found that the perturbed system could have at most six limit cycles with one class of the asymmetric perturbation and three limit cycles with the other class of asymmetric perturbation, which generalized the result of paper [12].

Appendix

The Proof of Lemma 5

Case 1 (). We have    and    for all    if  . Then,    for every  . Thus,

We now fix    and consider    which satisfies  . Then, we have

We now fix    and consider    which satisfies  . Then, we have

With simple computation, we get

where

Case 2 (). We have     and    for all    if  . Then,    for every  . Thus,

We fix now    and consider    which satisfies  . Then, we have

We now fix    and consider    which satisfies  . Obviously,  . Then, we have

With simple computation, we get

where

Case 3 (  and  ). We have    and    for all    if  .   Then,    for every  . Thus,

We fix now    and consider    which satisfies  . Then, we have

We now fix    and consider    which satisfies  . Then, we can write

With simple computation, we get

where

Case 4 (  and  ). We have    and    for all  if  . Then,    for every. Thus,

We fix now  and considerwhich satisfies. Then, we have

We now fixand consider    which satisfies  . Then, we have

With simple computation, we get

where

Case 5 (  and  ). We have    and    for all    if  .   Then,    for every  . Thus,

We fix now  and consider    which satisfies  . Then, we have

With simple computation, we get

where

This completes the proof of the lemma.

Acknowledgment

This work is supported in part by the National Natural Science Foundation of China (1097 2082).

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