# Bifurcation of Limit Cycles of a Class of Piecewise Linear Differential Systems inℜ4 with Three Zones

Authors: Yanyan Cheng
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## 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.


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## Body

## 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 ℜ4. 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 inℜ4  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(1)x˙=A0x+εF(x),
for ε≠0 sufficiently small real parameter, where
(2)A0=(0-1001000000-10010),
and F:ℜ4→ℜ4  is given by
(3)F(x)=Ax+φ(kTx)b,
with  A∈M4(ℜ),  k,b∈ℜ4∖{0},  and  φ:ℜ→ℜ  the piecewise linear function ifm1<0<m2,
(4)φ(x)={hm1,forx∈(-∞,m1),hx,forx∈(m1,m2),hm2,forx∈(m2,+∞);ifm1<m2<0,
(5)φ(x)={hm1-hm2,forx∈(-∞,m1),hx-hm2,forx∈(m1,m2),0,forx∈(m2,+∞);if0<m1<m2,
(6)φ(x)={0,forx∈(-∞,m1),hx-hm1,forx∈(m1,m2),hm2-hm1,forx∈(m2,+∞),whereh∈ℜ∖{0}. The independent variable is denoted by  t; vectors of  ℜ4  are column vectors, and  kT  denotes a transposed vector.Forε=0,  system (1) becomes
(7)x˙1=-x2,x˙2=x1,x˙3=-x4,x˙4=x3.Our main results are the following.Theorem 1.
Ifm1m2>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  ε  sufficiently small. Moreover, there are systems of form (1) having six limit cycles.Theorem 2.
Ifm1m2<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  ε  sufficiently small. Moreover, there are systems of form (1) having three limit cycles.It is worth to note that Theorem2 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.Theorems1 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:(8)x˙(t)=εH(t,ε)+ε2R(t,x,ε),
where  H:ℜ×D→ℜn,  R:ℜ×D×(-ε0,ε0)→ℜn  are continuous functions. T-periodic in the first variable, and  D  is an open subset of  ℜn. We define  h:D→ℜn  as
(9)h(z)=∫0TH(s,z)ds,
and assume that(i)
H and  R  are locally Lipschitz with respect to  x;(ii)
fora∈D  with  h(a)=0, there exists a neighborhood  V  of  a  such that  h(z)≠0  for all  z∈V-∖{a}  and  dB(h,V,0)≠0.
Then, for|ε|>0  sufficiently small, there exists an isolated T-periodic solution  ϕ(·,ε)  of system (8) such that  ϕ(·,ε)→a  as ε→0.We remind here thatdB(h,V,a)  denotes the Brouwer degree of the function  h  with respect to the set  V  and the point  a, as is defined in [19]. The following fact is useful for the proof of Theorems 1 and 2.Fact 1.
Leth:D→ℜn be a C1 function, with h(a)=0, where D is an open subset of ℜn and a∈D. Whenever a is a simple zero of  h (i.e., J(a)≠0), there exists a neighborhood V of a such that h(z)≠0 for all z∈V-∖{a}. Then, dB(h,V,0)∈{-1,1}.


## 3. Proof of Main Theorems

The proof of Theorems1 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  k  in the definition of the function  F(x)  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
(10)x˙=A0x+εA-x+εφ(x1)b-,
where  A-∈M4(ℜ)  is an arbitrary matrix and  b-=(b-1,b-2,0,0)T  or  b-=e3.Proof.
A linear change of variablesx=Py, with  P  invertible, transforms system (1) into
(11)y˙=P-1A0Py+εP-1APy+εφ(kTPy)P-1b,
where  b=(b1,b2,b3,b4)T,  k=(k1,k2,k3,k4)T.
We have to findP  invertible which satisfies
(12)P-1A0P=A0,kTP=e1T.
It is easy to obtain that  P-1  has the following form:
(13)P-1=(k1k2k3k4-k2k1-k4k3p31p32p33p34-p32p31-p34p33).
Thus, we have
(14)P-1b=b-,
where
(15)b-1=∑i=14kibi,b-2=-k2b1+k1b2-k4b3+k3b4,b-3=∑i=14p3ibi,b-4=-p32b1+p31b2-p34b3+p33b4.
If  b-12+b-22≠0, it is easy to find  P-1  invertible with  p31,  p32,  p33,  p34  satisfying
(16)b-3=∑i=14p3ibi=0,b-4=-p32b1+p31b2-p34b3+p33b4=0.
If  b-12+b-22=0, it is easy to find  P-1  invertible with  p31,  p32,  p33,  p34  satisfying
(17)b-3=∑i=14p3ibi=1,b-4=-p32b1+p31b2-p34b3+p33b4=0.
Changing  variables  y  to  x  with  x=y, then we obtain system (10).The standard form of the averaging method is obtained by changing variables(x1,x2,x3,x4) to (r,θ,ρ,s) with
(18)x1=rcosθ,x2=rsinθ,x3=ρcos(θ+s),x4=ρsin(θ+s).
Thus, system (10) is transformed into the following system:
(19)drdθ=εH1(θ,r,ρ,s)+ε2O(1),dρdθ=εH2(θ,r,ρ,s)+ε2O(1),dsdθ=εH3(θ,r,ρ,s)+ε2O(1),
where  H1,  H2,  and  H3  are given by
(20)H1=cosθF1+sinθF2,H2=cos(θ+s)F3+sin(θ+s)F4,H3=1rsinθF1-1rcosθF2-1ρsin(θ+s)F3+1ρcos(θ+s)F4
and for every  i=1,2,…,4,
(21)Fi=ai1rcosθ+ai2rsinθ+ai3ρcos(θ+s)+ai4ρsin(θ+s)+φ(rcosθ)b-i,
where  aij  are elements of the matrix  A-  of Lemma 4.We takeε0  sufficiently small,  m  arbitrarily large and
(22)Dm={(r,ρ,s)∣(r,ρ,s)∈(1m,m)2×S}.
Then, the vector of system (19) is well defined and continuous on  S×Dm×(-ε0,ε0). Moreover, the system is  2π-periodic with respect to variable  θ  and locally Lipschitz with respect to variables  (r,ρ,s). Our next step is to find the corresponding function  h:D→ℜ3,  h=(h1,h2,h3), where
(23)hi(r,ρ,s)=∫02πHi(r,θ,ρ,s)dθ,
for  i=1,2,3.In order to calculate the exact expression ofh, we denote
(24)I1(r)=∫02πφ(rcosθ)cosθdθ,(25)I2(r)=∫02πφ(rcosθ)sinθdθ,
for  each  r>0, where  φ  is the piecewise linear function given by (4)–(6). Without loss of generality, we assume that the slope  h  of  φ  is positive.Lemma 5.
The integralsI1  and  I2  given by (24)-(25), respectively, have the following expressions:
(26)I2(r)=0,∀r>0,m1,m2,
and(1)
if0<m1<m2,
(27)I1(r)={0,if0<r≤m1,J(r,m1),ifm1<r<m2,K(r,m1,m2),ifr≥m2;(2)
ifm1<m2<0,
(28)I1(r)={0,if0<r≤|m2|,-J(r,m2),if|m2|<r<|m1|,K(r,m1,m2),ifr≥|m1|;(3)
ifm1<0<m2  and  |m1|<|m2|,
(29)I1(r)={πhr,if0<r≤|m1|,πhr+J(r,m1),if|m1|<r<m2,πhr+K(r,m1,m2),ifr≥m2;(4)
ifm1<0<m2  and  |m1|>|m2|,
(30)I1(r)={πhr,if0<r≤m2,πhr-J(r,m2),ifm2<r<|m1|,πhr+K(r,m1,m2),ifr≥|m1|;(5)
ifm1<0<m2  and  |m1|=|m2|=m,
(31)I1(r)={πhr,if0<r≤m,πhr-2J(r,m),ifr>m,where(32)J(r,mi)=hr(arctanr2-mi2mi-mir2-mi2r2),fori=1,2, and
(33)K(r,m1,m2)=hr(arctanr2-m12m1-arctanr2-m22m2-m1r2-m12r2+m2r2-m22r2).The proof of this lemma is given in the appendix.Remark 6.
Ifm1<0<m2  and  |m1|=|m2|, system (1) can be transformed into the system which is studied in the paper [12].Lemma 7.
Ifm1m2>0, one defines  m-=max(|m1|,|m2|),  m_=min(|m1|,|m2|)  and consider the equation  I1(r)=cr, r>0  with  I1  given by (24), and  c is  a real parameter. Then, (1)
ifc<0  or  c>h(arctan(m-/m_)-arctan(m_/m-)), the equation has no solutions;(2)
ifc=0, then the interval  (0,m_]  is a continuum of solutions;(3)
ifc=h(arctan(m-/m_)-arctan(m_/m-)), there is an unique solution  r*=m-2+m_2;(4)
ifc∈(0,h(arctan(m-/m_)-arctan(m_/m-))), there are two solutions  r1*<m-2+m_2  and  r2*>m-2+m_2.Proof.
If0<m1<m2, we have  m-=m2  and  m_=m1. It is easy to see that all  r∈(0,m_]  are a solution if  c=0. If  c≠0  changing the variable  u=r2-m12/m1  and defining  g(u)=I1(r)/hr, we obtain the equivalent equation
(34)g(u)={arctanu-u1+u2,ifu∈(0,m22-m12m1],arctanu-u1+u2sfddddiiiiffsals-arctan(1+u2)m12-m22m2+m2(1+u2)m12-m22m12(1+u2),sfdifu∈(m22-m12m1,+∞)
with simple computation; we find that the function  g  is strictly monotonically increasing of variable  u  when  u∈(0,m2/m1)  and strictly monotonically decreasing when  u∈(m2/m1,+∞). The function  g  gets to the maximal value  gmax=arctan(m2/m1)-arctan(m1/m2)  when  u=m2/m1. Also we have  f(u)→0  as  u→0  and  u→+∞. The proof is similar if  m1<m2<0.Lemma 8.
Ifm1m2<0, one defines  m_=min(|m1|,|m2|)  and  consider the equation  I1(r)=cr,  r>0  with  I1  given by (24), and  c is  a real parameter. Then,(1)
ifc<0  or  c>πh, the equation has no solutions;(2)
ifc=πh, then the interval  (0,m_]  is a continuum of solutions;(3)
ifc∈(0,πh), there is an unique solution  r*.Proof.
We only consider the case when|m1|<|m2|: the proof is similar when  |m1|>|m2|  and  |m1|=|m2|. It is easy to see that all  r∈(0,|m1|]  are a solution if  c=πh. If  c≠0  changing the variable  u=r2-m12/m1  and defining  g(u)=I1(r)/hr, we obtain the equivalent equation
(35)g(u)={π+arctanu-u1+u2,ifu∈[m22-m12m1,0),π+arctanu-u1+u2sfdffsals-arctan(1+u2)m12-m22m2+m2(1+u2)m12-m22m12(1+u2),sfdifu∈(-∞,m22-m12m1).
With simple computation, we find that the function  g  is strictly monotonically increasing of variable  u. It is easy to know  f(u)→π  as  u→0  and  f(u)→0  as  u→-∞.With Lemma5, we obtain the expressions for the components of function  h,
(36)h1(r,ρ,s)=c1r+c2ρcoss+c3ρsins+b-1I1(r),h2(r,ρ,s)=c7ρ+c5rcoss+c6rsins+b-3cossI1(r),h3(r,ρ,s)=c4+(c3coss-c2sins)ρr+(c6coss-c5sins)rρ-b-3sinsI1(r)ρ-b-2I1(r)r,
where  ci  are constants that depend linearly on  aij(37)c1=(a11+a22)π,c2=(a13+a24)π,c3=(a14-a23)π,c4=(a43+a12-a34-a21)π,c5=(a31+a42)π,c6=(a41-a32)π,c7=(a33+a44)π.
According to Theorem 3 and Fact 1, for each simple zero  (r*,ρ*,s*)  of (36) there is an isolated  2π-periodic solution    ϕ(·,ε)  of system (19) with  |ε|≠0  sufficiently small such that  ϕ(·,ε)→(r*,ρ*,s*)  as ε→0. Any isolated  2π-periodic solution of system (19) with  |ε|≠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
(38)I1(r)=k2(s)d(s)ρ,r=k1(s)d(s)ρ,
where
(39)d(s)=(b-1c5-b-3c1)coss+b-1c6sins,k1(s)=b-3coss(c2coss+c3sins)+b-1c7,k2(s)=c1c7-c2c5cos2s-c3c6sin2s-(c2c6+c3c5)sinscoss.
Substituting (38) into the third equation, we obtain
(40)h3(r,ρ,s)=f(s)d(s)k1(s)=0,
where
(41)f(s)=c4d(s)k1(s)+(c3cos-c2sins)d2(s)+(c6coss-c5sins)k12(s)-b-3sinsk1(s)k2(s)-b-2d(s)k2(s).
It is necessary to study the zeros of  f  instead of the zeros of  h.Lemma 9.
The functionf:[0,2π)→ℜ  given by formula (41) can have at most six isolated zeros, and they appear in pairs  {s*,s*+π(mod2π)}.Proof.
Substitutingcoss=x  and  sins=1-x2  in  f(s)=0  we get
(42)D1x+D3x3+(D0+D2x2)1-x2=0,
where
(43)D0=b-12(c2c62-c4c6c7+c5c72)+b-1b-2(c3c62-c1c6c7),D1=b-12(2c2c5c6-c3c62-c4c5c7-c6c72)+b-32(c1c3c7-c6c32)+b-1b-2(2c2c5c6+c3c52-c1c5c7),D2=b-12(c2c52-c2c62-2c3c5c6)+b-32(c1c2c7+c2c12-c1c3c4-2c2c3c6)+b-1b-2(2c2c5c6+c3c52-c3c62),D3=b-12(c3c62-c3c52-2c2c5c6)+b-32(c32c6-c3c12-c1c2c4-c1c3c7-c6c22)+b-1b-2(c2c52-2c3c5c6-c2c62).
When we consider the case  coss=x  and  sins=-1-x2,  f(s)=0  becomes
(44)D1x+D3x3-(D0+D2x2)1-x2=0.
It follows that we have to find solutions of (42) or (44) in the interval  [-1,1]. This is equivalent to
(45)D1x+D3x3-(D0+D2x2)2(1-x2)=0
which is the polynomial equation
(46)(D32+D22)x6+(2D1D3+2D0D2-D22)x4+(D12+D02-2D0D2)x2-D02=0.
This equation can have at most six roots in the interval  [-1,1]. Then,  f(s)=0  has at most six solutions  s∈[0,2π).   Since  f(s+π)=-f(s)  for all  s∈[0,2π), it is clear that if  s*  is a zero of  f  then  s*+π(mod2π)  is also a zero.The functionsf(s),  d(s),  k1(s),  and  k2(s)  have the properties  f(s+π)=-f(s),  d(s+π)=-d(s),  k1(s+π)=k1(s),  and  k2(s+π)=k2(s). So, we have
(47)k1(s)d(s)>0⟹k1(s+π)d(s+π)<0.
Thus, the equation h3=0  at most three zeros that satisfy  k1(s)/d(s)>0. With Lemma 7 for a fixed  s*, we at most find two isolated value of  r*  from  I1(r)/r=k2(s*)/d(s*). With Lemma 8 for a fixed  s*, we at most find one isolated value of  r*  from  I1(r)/r=k2(s*)/d(s*). For fixed  s*  and fixed  r*,  ρ/r=k1(s*)/d(s*)  gives at most one isolated value for  ρ*. Thus, we conclude that if  m1m2>0  the maximum number of limit cycles for system (1) is six, and if  m1m2<0  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  (r*,ρ*,s*)  is given by the eigenvalues of the matrix
(48)∂(h1,h2,h3)∂(r,ρ,s)|(r,ρ,s)=(r*,ρ*,s*).
In order to show that there exist examples with exactly six limit cycles, we consider the following values of the coefficients:
(49)c1=-1,c2=-433,c3=0,c4=-433,c5=0,c6=1,c7=-1,b-1=1,b-2=b-3=b-4=0,h=6,m1=1,m2=3.
More precisely, the system has the following form:
(50)x˙=A0x+εAx+εφ(x1)b-,
where
(51)φ(x1)={0,forx1∈(-∞,1),6x1-6,forx1∈(1,3),63-6,forx1∈(3,+∞),A0=(0-1001000000-10010),A=(a11a12a13a14a21a22a14a24a31a32a33a34a41-a31a43a44),b-=(1000),
and  aij∈ℜ  satisfy
(52)(a11+a22)π=-1,(a13+a24)π=-433,(a43+a12-a34-a21)π=-433,(a41-a32)π=1,(a33+a44)π=-1.
It is easy to know6(arctan3-arctan(3/3))=π. Computing the six solutions of  f(s)=0, we get  {π/6,π/3,π/2,7π/6,4π/3,3π/2}. The values of  d(s*),  k1(s*)  and  k2(s*)  are given in Table 1.Table 1
The values ofs*, d(s*), k1(s*), and k2(s*).
s
*
k
1
(
s
*
)
d
(
s
*
)
k
2
(
s
*
)
π
/
6
1
1
/
2
2
π
/
3
1
3
/
2
2
π
/
2
1
1
1
7
π
/
6
1
-
1
/
2
2
4
π
/
3
1
-
3
/
2
2
3
π
/
2
1
-
1
1There are three values ofs*  that satisfy  k1(s*)/d(s*)>0  and  0<k2(s*)/k1(s*)<π. These three solutions are  {π/6,π/3,π/2}.The six values of solutions*,  r*,  ρ*  and the value of the Jacobian at the solution  (r*,ρ*,s*)  are given in Table 2.Table 2
The values of solutions*,r*,ρ* and the Jacobian Jh(r*,ρ*,s*).
s
*
r
*
ρ
*
J
h
(
r
*
,
ρ
*
,
s
*
)
π
/
6
1.484
0.742
11.957
π
/
6
4.139
2.07
-
4.688
π
/
3
1.484
1.285
-
3.27
π
/
3
4.139
3.585
-
1.144
π
/
2
1.254
1.254
5.732
π
/
2
8.672
8.672
-
0.964


## 4. Conclusion

In this paper, we have studied the limit cycles bifurcated from periodic orbits of a linear differential system inℜ4  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].



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