- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Discrete Dynamics in Nature and Society

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

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

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

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.

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.

#### 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).

#### References

- J. Llibre and M. A. Texeira, “On the stable limit cycle of a weight-driven pendulum clock,”
*European Journal of Physics*, vol. 31, no. 1, pp. 249–1254, 2010. View at Publisher · View at Google Scholar - M. Vidyasagar,
*Nonlinear Systems Analysis*, Poelltice-Hall, 2nd edition, 1993. - M. di Bernardo, C. J. Budd, A. R. Champneys, and P. Kowalczyk,
*Piecewise-Smooth Dynamical Systems*, vol. 163 of*Applied Mathematical Sciences*, Springer, London, UK, 2008, Theory and applications. View at Zentralblatt MATH · View at MathSciNet - V. Carmona, E. Freire, E. Ponce, and F. Torres, “Bifurcation of invariant cones in piecewise linear homogeneous systems,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 15, no. 8, pp. 2469–2484, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Huan and X.-S. Yang, “Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 75, no. 17, pp. 6260–6274, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Chritopher and C. Li,
*Limit Cycles in Differential Equations*, Birkhauser, Boston, Mass, USA, 2007. - S.-M. Huan and X.-S. Yang, “On the number of limit cycles in general planar piecewise linear systems,”
*Discrete and Continuous Dynamical Systems A*, vol. 32, no. 6, pp. 2147–2164, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Li, “Hilbert's 16th problem and bifurcations of planar polynomial vector fields,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 13, no. 1, pp. 47–106, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Llibre and A. Makhlouf, “Bifurcation of limit cycles from a two-dimensional center inside ${\Re}^{4}$,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1387–1392, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Freire, E. Ponce, and J. Ros, “The focus-center-limit cycle bifurcation in symmetric 3D piecewise linear systems,”
*SIAM Journal on Applied Mathematics*, vol. 65, no. 6, pp. 1933–1951, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Carmona, S. Fernández-García, F. Fernández-Sánchez, E. García-Medina, and A. E. Teruel, “Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 75, no. 15, pp. 5866–5883, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Buică and J. Llibre, “Bifurcation of limit cycles from a four-dimensional center in control systems,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 15, no. 8, pp. 2653–2662, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Buzzi, J. Llibre, and J. C. Medrado, “On the limit cycles of a class of piecewise linear differential systems in ${\Re}^{4}$ with two zones,”
*Mathematics and Computers in Simulation*, vol. 82, no. 4, pp. 533–539, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Buzzi, J. Llibre, J. C. Medrado, and J. Torregrosa, “Bifurcation of limit cycles from a centre in ${\Re}^{4}$ in resonance 1:
*N*,”*Dynamical Systems*, vol. 24, no. 1, pp. 123–137, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. T. Cardin, T. de Carvalho, and J. Llibre, “Bifurcation of limit cycles from an $n$-dimensional linear center inside a class of piecewise linear differential systems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 75, no. 1, pp. 143–152, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - J. Llibre, E. Ponce, and J. Ros, “Algebraic determination of limit cycles in a family of three-dimensional piecewise linear differential systems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, no. 17, pp. 6712–6727, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Buică and J. Llibre, “Averaging methods for finding periodic orbits via Brouwer degree,”
*Bulletin des Sciences Mathématiques*, vol. 128, no. 1, pp. 7–22, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Buică, J. Llibre, and O. Makarenkov, “Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator,”
*SIAM Journal on Mathematical Analysis*, vol. 40, no. 6, pp. 2478–2495, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. E. Browder, “Fixed point theory and nonlinear problems,”
*American Mathematical Society*, vol. 9, no. 1, pp. 1–39, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet