Bifurcation of Limit Cycles of a Class of Piecewise Linear Differential Systems in R 4 with Three Zones

We study the bifurcation of limit cycles from periodic orbits of a four-dimensional systemwhen 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.


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][7][8][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 highdimensional cases, there are a few papers [12][13][14][15][16]. Especially in [12] the authors studied the bifurcation of limit cycles of a class of piecewise linear systems in R 4 .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 R 4 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  ̸ = 0 sufficiently small real parameter, where and  : R 4 → R 4 is given by with  ∈  4 (R), ,  ∈ R 4 \ {0}, and  : R → R the piecewise linear function for  ∈ ( 2 , +∞) ; (5) where ℎ ∈ R \ {0}.The independent variable is denoted by ; vectors of R 4 are column vectors, and   denotes a transposed vector.
For  = 0, system (1) becomes Our main results are the following.

Theorem 1.
If  1  2 > 0, 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  1  2 < 0, 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 Buicȃ 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.

First-Order Averaging Method
The aim of this section is to review the first-order averaging method which is obtained by Buicȃ and Llibre [17].The advantage of this method is that the smoothness assumptions for the vector field of the differential system are minimal.
We remind here that   (ℎ, , ) denotes the Brouwer degree of the function ℎ 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.

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.
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  −1 has the following form: Thus, we have where If Changing variables  to  with  = , then we obtain system (10).
The standard form of the averaging method is obtained by changing variables ( 1 ,  2 ,  3 ,  4 ) to (, , , ) with Thus, system ( 10) is transformed into the following system: where  1 ,  2 , and  3 are given by and for every  = 1, 2, . . ., 4, where   are elements of the matrix  of Lemma 4.
In order to calculate the exact expression of ℎ, we denote for each  > 0, where  is the piecewise linear function given by ( 4)- (6).Without loss of generality, we assume that the slope ℎ of  is positive.
Lemma 5.The integrals  1 and  2 given by (24)-( 25), respectively, have the following expressions: and where for  = 1, 2, and The proof of this lemma is given in the appendix.
can be transformed into the system which is studied in the paper [12].
Proof.We only consider the case when | ) . ( With simple computation, we find that the function  is strictly monotonically increasing of variable .It is easy to know () →  as  → 0 and () → 0 as  → −∞.
With Lemma 5, we obtain the expressions for the components of function ℎ, where According to Theorem 3 and Fact 1, for each simple zero ( * ,  * ,  * ) of (36) there is an isolated 2-periodic solution (⋅, ) of system (19) with || ̸ = 0 sufficiently small such that (⋅, ) → ( * ,  * ,  * ) as  → 0. Any isolated 2-periodic solution of system (19) with || ̸ = 0 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 ℎ.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 ℎ.
With Lemma 8 for a fixed  * , we at most find one isolated value of  * from  1 ()/ =  2 ( * )/( * ).For fixed  * and fixed  * , / =  1 ( * )/( * ) gives at most one isolated value for  * .Thus, we conclude that if  1  2 > 0 the maximum number of limit cycles for system (1) is six, and if  1  2 < 0 the maximum number of limit cycles for system (1) is three.
The six values of solution  * ,  * ,  * and the value of the Jacobian at the solution ( * ,  * ,  * ) are given in Table 2.This completes the proof of the lemma.