# On the Limit Cycles for a Class of Perturbed Fifth-Order Autonomous Differential Equations

Authors: Nabil Sellami; Romaissa Mellal; Bahri Belkacem Cherif; Sahar Ahmed Idris
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

We study the limit cycles of the fifth-order differential equationx⋅⋅⋅⋅⋅−ex⃜−dx⃛−cx¨−bx˙−ax=εFx,x˙,x¨,x⋯,x⃜ with a=λμδ,b=−λμ+λδ+μδ,c=λ+μ+δ+λμδ,d=−1+λμ+λδ+μδ,e=λ+μ+δ, where ε is a small enough real parameter, λ,μ, and δ are real parameters, and F∈C2 is a nonlinear function. Using the averaging theory of first order, we provide sufficient conditions for the existence of limit cycles of this equation.


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## Body

## 1. Introduction and Statement of the Main Results

The study of the limit cycles is one of the main topics of the qualitative theory of differential equations and dynamical systems. A limit cycle of a differential equation is an isolated periodic orbit of this equation; it means that there is no periodic orbits in the vicinity of this limit cycle. There are several theories and methods for the study of the existence, uniqueness, or number and stability of limit cycles of differential equations which have been developed in trying to answer Hilbert’s sixteenth problem posed in 1900 [1] about the maximum number of limit cycles that a planar polynomial differential system can have.The averaging theory is one of the most important tools used actually to the study of limit cycles for second and higher order differential equations, you can see in [2–8]. More details on the averaging theory can be found in the books of Sanders and Verhulst [9] and of Verhulst [9].In [7], the authors studied the limit cycles of the following third-order differential equation(1)x⃛−μx¨+x˙−μx=εFx,x˙,x¨,t,with μ≠0; ε is a small real parameter; F∈C2 is 2π− periodic in t.In [5], the authors studied equation (1) with F=Fx,x˙,x¨ which is autonomous. They studied the two cases μ≠0 and μ=0.In [6], the authors studied the following fourth-order differential equation:(2)x⃜−λ+μx⃛+1+λμx¨−λ+μx˙+λμx=εFx,x˙,x¨,x⃛,t,where λ and μ are real, ε is a small real parameter, and F∈C2 is 2π− periodic in t.In [4], the authors studied equation (2) with F=Fx,x˙,x¨,x⃛ which is autonomous.In this paper, we shall use a result of the averaging theory to study the limit cycles of the following class of fifth-order autonomous ordinary differential equations:(3)x⋅⋅⋅⋅⋅−ex⃜−dx⃛−cx¨−bx˙−ax=εFx,x˙,x¨,x⋯,x⃜,where(4)a=λμδ,b=−λμ+λδ+μδ,c=λ+μ+δ+λμδ,d=−1+λμ+λδ+μδ,e=λ+μ+δ,where the dot means derivative with respect to an independent variable t, ε is a small enough parameter, and F∈C2 is a nonlinear function. Here, the variable x and the parameters λ,μ,δ and ε are real.In [8], the authors studied equation (3) with F=Fx,x˙,x¨,x⃛,x⃜,t which depends explicitly on the independent variable t. Here, we study the autonomous case using a different approach. Note that our results are distinct and new.Now, we state our main results for the limit cycles of equation (3).For the different values of the parametersλ,μ, and δ, we distinguish the five following cases.Case 1.
λμδ≠0 and λ≠μ≠δ.Case 2.
λ=0,μδ≠0, and μ≠δ.Case 3.
λ=0 and μ=δ≠0.Case 4.
λ≠0 and μ=δ≠0.Case 5.
λ=μ=δ≠0.
For each one of these cases, we will give a theorem which provides sufficient conditions for the existence of limit cycles of equation (3) and we provide also an application.
There are two other cases (λ=μ=0 and δ≠0) and λ=μ=δ=0 that we cannot study because they are too much degenerated for Theorem 6.


### 1.1. Case1: λμδ≠0 and λ≠μ≠δ

In order to state our results for this case, we define the function(5)ℱr0=12π02πsinθFA1,A2,A3,A4,A5dθ,where(6)A1=λμδ−λ−μ−δcosθ+1−λμ−λδ−μδsinθr01+λ21+μ21+δ2,A2=1−λμ−λδ−μδcosθ+λ+μ+δ−λμδsinθr01+λ21+μ21+δ2,A3=λ+μ+δ−λμδcosθ+λμ+λδ+μδ−1sinθr01+λ21+μ21+δ2,A4=λμ+λδ+μδ−1cosθ+λμδ−λ−μ−δsinθr01+λ21+μ21+δ2,A5=λμδ−λ−μ−δcosθ+1−λμ−λδ−μδsinθr01+λ21+μ21+δ2.Our main result for this case is the following theorem.Theorem 1.
Assume thatλμδ≠0 and λ≠μ≠δ. For every positive simple zero r0∗ of the function ℱr0 given by (5) there is a limit cycle xt,ε of equation (3) tending to the periodic solution(7)x∗t=λμδ−λ−μ−δcost+1−λμ−λδ−μδsintr0∗1+λ21+μ21+δ2,of(8)x⋅⋅⋅⋅⋅−λ+μ+δx⃜+1+λμ+λδ+μδx⃛−λ+μ+δ+λμδx¨+λμ+λδ+μδx˙−λμδx=0,when ε⟶0.
Theorem1 will be proved in Section 3.1.1.
An application of Theorem1 is the following.Corollary 1.
Assume thatλμδ≠0, λ≠μ≠δ, λμ+λδ+μδ≠1, and(9)Fx,x˙,x¨,x⃛,x⃜=x5−x˙4−x¨3−x⃛2−x⃜−1.
Then, there is a limit cyclex1t,ε of equation (3) tending to the periodic solution(10)x1∗t=2λμδ−λ−μ−δcost+1−λμ−λδ−μδsint51+λ21+μ21+δ2,of equation (8) when ε⟶0.
Corollary1 will be proved in Section 3.1.2.


### 1.2. Case2: λ=0,μδ≠0, and μ≠δ

We define the functions(11)ℱ1r0,Z0=12π02πsinθFB1,B2,B3,B4,B5dθ,ℱ2r0,Z0=12π02πFB1,B2,B3,B4,B5dθ,and(12)B1=−μ−δcosθ+1−μδsinθr01+μ21+δ2+Z0μδ,B2=1−μδcosθ+μ+δsinθr01+μ21+δ2,B3=μ+δcosθ+μδ−1sinθr01+μ21+δ2,B4=μδ−1cosθ−μ+δsinθr01+μ21+δ2,B5=−μ+δcosθ+1−μδsinθr01+μ21+δ2.Our main result for this case is the following theorem.Theorem 2.
Assume thatλ=0,μδ≠0, and μ≠δ. For every zero r0∗,Z0∗ of the system ℱ1r0,Z0=ℱ2r0,Z0=0 where ℱ1 and ℱ2 are given by (11) such that(13)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗≠0,there is a limit cycle xt,ε of equation (3) tending to the periodic solution(14)x∗t=−μ−δcost+1−μδsintr0∗1+μ21+δ2+Z0∗μδ,of(15)x⋅⋅⋅⋅⋅−μ+δx⃜+1+μδx⃛−μ+δx¨+μδx˙=0,when ε⟶0.
Theorem2 will be proved in Section 3.2.1.
An application of Theorem2 is the following.Corollary 2.
Assume thatλ=0,μδ≠0, μ≠δ, μδ≠1, and Fx,x˙,x¨,x⃛,x⃜=x2−2x+x˙2, then there is a limit cycle x2t,ε of equation (3) tending to the periodic solution(16)x2∗t=−μ−δcost+1−μδsint1+μ21+δ2+1,of equation (15) when ε⟶0.
Corollary2 will be proved in Section 3.


### 1.3. Case3: λ=0 and μ=δ≠0

We define the functions(17)ℱ1r0,Z0=12π02π−sinθFC1,C2,C3,C4,C5dθ,ℱ2r0,Z0=12π02π12π02πFC1,C2,C3,C4,C5dθ,where(18)C1=2μcosθ+μ2−1sinθr01+μ22+Z0μ2,C2=μ2−1cosθ−2μsinθr01+μ22,C3=−2μcosθ−μ2−1sinθr01+μ22,C4=−μ2−1cosθ+2μsinθr01+μ22,C5=2μcosθ+μ2−1sinθr01+μ22.Our main result for this case is the following theorem.Theorem 3.
Assume thatλ=0 and μ=δ≠0. For every zero r0∗,Z0∗ of the system ℱ1r0,Z0=ℱ2r0,Z0=0 where ℱ1 and ℱ2 are given by (17) such that(19)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗≠0,there is a limit cycle xt,ε of equation (3) tending to the periodic solution(20)x∗t=2μcost+μ2−1sintr0∗1+μ22+Z0∗μ2,of(21)x⋅⋅⋅⋅⋅−2μx⃜+1+μ2x⃛−2μx¨+μ2x˙=0,when ε⟶0.
Theorem3 will be proved in Section 3.3.1.
An application of Theorem3 is the following.Corollary 3.
Assume thatλ=0, δ=μ≠0,μ≠−1±5/2, and Fx,x˙,x¨,x⃛,x⃜=x2−xx˙−1, then there is a limit cycle x3t,ε of equation (3) tending to the periodic solution(22)x3∗t=2μcost+μ2−1sint21+μ2,of equation (21) when ε⟶0.
Corollary3 will be proved in Section 3.3.2.


### 1.4. Case4: λ≠0 and μ=δ≠0

We define the function(23)ℱr0=12π02π−sinθFD1,D2,D3,D4,D5dθ,where(24)D1=λ+2μ−λμ2cosθ+μ2+2λμ−1sinθr01+λ21+μ22,D2=μ2+2λμ−1cosθ−λ+2μ−λμ2sinθr01+λ21+μ22,D3=−λ+2μ−λμ2cosθ−μ2+2λμ−1sinθr01+λ21+μ22,D4=−μ2+2λμ−1cosθ+λ+2μ−λμ2sinθr01+λ21+μ22,D5=λ+2μ−λμ2cosθ+μ2+2λμ−1sinθr01+λ21+μ22.Our main result for this case is the following theorem.Theorem 4.
Assume thatλ≠0 and δ=μ≠0. For every positive simple zero r0∗ of the function ℱr0 given by (23), there is a limit cycle xt,ε of equation (3) tending to the periodic solution(25)x∗t=λ+2μ−λμ2cost+μ2+2λμ−1sintr0∗1+λ21+μ22,of(26)x⋅⋅⋅⋅⋅−λ+2μx⃜+1+2λμ+μ2x⃛−λ+2μ+λμ2x¨+2λμ+μ2x˙−λμ2x=0,when ε⟶0.
Theorem4 will be proved in Section 3.4.1.
An application of Theorem4 is the following.Corollary 4.
Assume thatλ≠0, δ=μ≠0, μ2+2λμ−1≠0, and Fx,x˙,x¨,x⃛,x⃜=x3−x−1, then there is a limit cycle x4t,ε of equation (3) tending to the periodic solution(27)x4∗t=23λ+2μ−λμ2cost+μ2+2λμ−1sint31+λ21+μ2,of equation (26) when ε⟶0.
Corollary4 will be proved in Section 3.4.2.


### 1.5. Case5: λ=μ=δ≠0

In order to state our result for this case, we define the function(28)ℱr0=12π02πsinθFE1,E2,E3,E4,E5dθ,where(29)E1=λ3−3λcosθ+1−3λ2sinθr01+λ23,E2=1−3λ2cosθ−λ3−3λsinθr01+λ23,E3=−λ3−3λcosθ−1−3λ2sinθr01+λ23,E4=−1−3λ2cosθ+λ3−3λsinθr01+λ23,E5=λ3−3λcosθ+1−3λ2sinθr01+λ23.Our main result for this case is the following theorem.Theorem 5.
Assume thatμ=δ=λ≠0. For every positive simple zero r0∗ of the function ℱr0 given by (28), there is a limit cycle xt,ε of equation (3) tending to the periodic solution(30)x∗t=λ3−3λcost+1−3λ2sintr0∗1+λ23,of(31)x⋅⋅⋅⋅⋅−3λx⃜+1+3λ2x⃛−3λ+λ3x¨+3λ2x˙−λ3x=0,when ε⟶0.
Theorem5 will be proved in Section 3.5.1.
An application of Theorem5 is the following.Corollary 5.
Assume thatμ=δ=λ≠0, λ≠±3, λ≠±3/3, and Fx,x˙,x¨,x⃛,x⃜=−x¨2x⃜2+x3+x2−x˙x¨+x⃛−1, then there is a limit cycle x5t,ε of equation (3) tending to the periodic solution(32)x5∗t=233λ2−1λλ4−2λ2−3λ3−3λcost+1−3λ2sint33λ2−11+λ22,of equation (31) when ε⟶0.
Corollary5 will be proved in Section 3.5.2.


## 1.1. Case1: λμδ≠0 and λ≠μ≠δ

In order to state our results for this case, we define the function(5)ℱr0=12π02πsinθFA1,A2,A3,A4,A5dθ,where(6)A1=λμδ−λ−μ−δcosθ+1−λμ−λδ−μδsinθr01+λ21+μ21+δ2,A2=1−λμ−λδ−μδcosθ+λ+μ+δ−λμδsinθr01+λ21+μ21+δ2,A3=λ+μ+δ−λμδcosθ+λμ+λδ+μδ−1sinθr01+λ21+μ21+δ2,A4=λμ+λδ+μδ−1cosθ+λμδ−λ−μ−δsinθr01+λ21+μ21+δ2,A5=λμδ−λ−μ−δcosθ+1−λμ−λδ−μδsinθr01+λ21+μ21+δ2.Our main result for this case is the following theorem.Theorem 1.
Assume thatλμδ≠0 and λ≠μ≠δ. For every positive simple zero r0∗ of the function ℱr0 given by (5) there is a limit cycle xt,ε of equation (3) tending to the periodic solution(7)x∗t=λμδ−λ−μ−δcost+1−λμ−λδ−μδsintr0∗1+λ21+μ21+δ2,of(8)x⋅⋅⋅⋅⋅−λ+μ+δx⃜+1+λμ+λδ+μδx⃛−λ+μ+δ+λμδx¨+λμ+λδ+μδx˙−λμδx=0,when ε⟶0.
Theorem1 will be proved in Section 3.1.1.
An application of Theorem1 is the following.Corollary 1.
Assume thatλμδ≠0, λ≠μ≠δ, λμ+λδ+μδ≠1, and(9)Fx,x˙,x¨,x⃛,x⃜=x5−x˙4−x¨3−x⃛2−x⃜−1.
Then, there is a limit cyclex1t,ε of equation (3) tending to the periodic solution(10)x1∗t=2λμδ−λ−μ−δcost+1−λμ−λδ−μδsint51+λ21+μ21+δ2,of equation (8) when ε⟶0.
Corollary1 will be proved in Section 3.1.2.


## 1.2. Case2: λ=0,μδ≠0, and μ≠δ

We define the functions(11)ℱ1r0,Z0=12π02πsinθFB1,B2,B3,B4,B5dθ,ℱ2r0,Z0=12π02πFB1,B2,B3,B4,B5dθ,and(12)B1=−μ−δcosθ+1−μδsinθr01+μ21+δ2+Z0μδ,B2=1−μδcosθ+μ+δsinθr01+μ21+δ2,B3=μ+δcosθ+μδ−1sinθr01+μ21+δ2,B4=μδ−1cosθ−μ+δsinθr01+μ21+δ2,B5=−μ+δcosθ+1−μδsinθr01+μ21+δ2.Our main result for this case is the following theorem.Theorem 2.
Assume thatλ=0,μδ≠0, and μ≠δ. For every zero r0∗,Z0∗ of the system ℱ1r0,Z0=ℱ2r0,Z0=0 where ℱ1 and ℱ2 are given by (11) such that(13)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗≠0,there is a limit cycle xt,ε of equation (3) tending to the periodic solution(14)x∗t=−μ−δcost+1−μδsintr0∗1+μ21+δ2+Z0∗μδ,of(15)x⋅⋅⋅⋅⋅−μ+δx⃜+1+μδx⃛−μ+δx¨+μδx˙=0,when ε⟶0.
Theorem2 will be proved in Section 3.2.1.
An application of Theorem2 is the following.Corollary 2.
Assume thatλ=0,μδ≠0, μ≠δ, μδ≠1, and Fx,x˙,x¨,x⃛,x⃜=x2−2x+x˙2, then there is a limit cycle x2t,ε of equation (3) tending to the periodic solution(16)x2∗t=−μ−δcost+1−μδsint1+μ21+δ2+1,of equation (15) when ε⟶0.
Corollary2 will be proved in Section 3.


## 1.3. Case3: λ=0 and μ=δ≠0

We define the functions(17)ℱ1r0,Z0=12π02π−sinθFC1,C2,C3,C4,C5dθ,ℱ2r0,Z0=12π02π12π02πFC1,C2,C3,C4,C5dθ,where(18)C1=2μcosθ+μ2−1sinθr01+μ22+Z0μ2,C2=μ2−1cosθ−2μsinθr01+μ22,C3=−2μcosθ−μ2−1sinθr01+μ22,C4=−μ2−1cosθ+2μsinθr01+μ22,C5=2μcosθ+μ2−1sinθr01+μ22.Our main result for this case is the following theorem.Theorem 3.
Assume thatλ=0 and μ=δ≠0. For every zero r0∗,Z0∗ of the system ℱ1r0,Z0=ℱ2r0,Z0=0 where ℱ1 and ℱ2 are given by (17) such that(19)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗≠0,there is a limit cycle xt,ε of equation (3) tending to the periodic solution(20)x∗t=2μcost+μ2−1sintr0∗1+μ22+Z0∗μ2,of(21)x⋅⋅⋅⋅⋅−2μx⃜+1+μ2x⃛−2μx¨+μ2x˙=0,when ε⟶0.
Theorem3 will be proved in Section 3.3.1.
An application of Theorem3 is the following.Corollary 3.
Assume thatλ=0, δ=μ≠0,μ≠−1±5/2, and Fx,x˙,x¨,x⃛,x⃜=x2−xx˙−1, then there is a limit cycle x3t,ε of equation (3) tending to the periodic solution(22)x3∗t=2μcost+μ2−1sint21+μ2,of equation (21) when ε⟶0.
Corollary3 will be proved in Section 3.3.2.


## 1.4. Case4: λ≠0 and μ=δ≠0

We define the function(23)ℱr0=12π02π−sinθFD1,D2,D3,D4,D5dθ,where(24)D1=λ+2μ−λμ2cosθ+μ2+2λμ−1sinθr01+λ21+μ22,D2=μ2+2λμ−1cosθ−λ+2μ−λμ2sinθr01+λ21+μ22,D3=−λ+2μ−λμ2cosθ−μ2+2λμ−1sinθr01+λ21+μ22,D4=−μ2+2λμ−1cosθ+λ+2μ−λμ2sinθr01+λ21+μ22,D5=λ+2μ−λμ2cosθ+μ2+2λμ−1sinθr01+λ21+μ22.Our main result for this case is the following theorem.Theorem 4.
Assume thatλ≠0 and δ=μ≠0. For every positive simple zero r0∗ of the function ℱr0 given by (23), there is a limit cycle xt,ε of equation (3) tending to the periodic solution(25)x∗t=λ+2μ−λμ2cost+μ2+2λμ−1sintr0∗1+λ21+μ22,of(26)x⋅⋅⋅⋅⋅−λ+2μx⃜+1+2λμ+μ2x⃛−λ+2μ+λμ2x¨+2λμ+μ2x˙−λμ2x=0,when ε⟶0.
Theorem4 will be proved in Section 3.4.1.
An application of Theorem4 is the following.Corollary 4.
Assume thatλ≠0, δ=μ≠0, μ2+2λμ−1≠0, and Fx,x˙,x¨,x⃛,x⃜=x3−x−1, then there is a limit cycle x4t,ε of equation (3) tending to the periodic solution(27)x4∗t=23λ+2μ−λμ2cost+μ2+2λμ−1sint31+λ21+μ2,of equation (26) when ε⟶0.
Corollary4 will be proved in Section 3.4.2.


## 1.5. Case5: λ=μ=δ≠0

In order to state our result for this case, we define the function(28)ℱr0=12π02πsinθFE1,E2,E3,E4,E5dθ,where(29)E1=λ3−3λcosθ+1−3λ2sinθr01+λ23,E2=1−3λ2cosθ−λ3−3λsinθr01+λ23,E3=−λ3−3λcosθ−1−3λ2sinθr01+λ23,E4=−1−3λ2cosθ+λ3−3λsinθr01+λ23,E5=λ3−3λcosθ+1−3λ2sinθr01+λ23.Our main result for this case is the following theorem.Theorem 5.
Assume thatμ=δ=λ≠0. For every positive simple zero r0∗ of the function ℱr0 given by (28), there is a limit cycle xt,ε of equation (3) tending to the periodic solution(30)x∗t=λ3−3λcost+1−3λ2sintr0∗1+λ23,of(31)x⋅⋅⋅⋅⋅−3λx⃜+1+3λ2x⃛−3λ+λ3x¨+3λ2x˙−λ3x=0,when ε⟶0.
Theorem5 will be proved in Section 3.5.1.
An application of Theorem5 is the following.Corollary 5.
Assume thatμ=δ=λ≠0, λ≠±3, λ≠±3/3, and Fx,x˙,x¨,x⃛,x⃜=−x¨2x⃜2+x3+x2−x˙x¨+x⃛−1, then there is a limit cycle x5t,ε of equation (3) tending to the periodic solution(32)x5∗t=233λ2−1λλ4−2λ2−3λ3−3λcost+1−3λ2sint33λ2−11+λ22,of equation (31) when ε⟶0.
Corollary5 will be proved in Section 3.5.2.


## 2. The Main Tool (First-Order Averaging Theory)

In this section, we present the basic result from the averaging theory that we need for proving the main results of this article.We consider the problem of the bifurcation ofT−periodic solutions from the differential system(33)x˙t=F0x,t+εF1x,t+ε2F2x,t,ε,with ε=0 to ε≠0 sufficiently small. The functions F0,F1:Ω×ℝ⟶ℝn and F2:Ω×ℝ×−ε0,ε0⟶ℝn are C2 functions, T−periodic is in the variable t, and Ω is an open subset of ℝn. We suppose that the unperturbed system(34)x˙t=F0x,t,has a k-dimensional submanifold Z of periodic solutions.Letxt,z be the solution of the unperturbed system (34) such that x0,z=z. The linearisation of system (34) along the periodic solution xt,z is written as(35)y˙=DxF0xt,z,ty.We denote byMzt some fundamental matrices of the linear differential system (35) and by ξ:ℝk×ℝn−k⟶ℝk the projection of ℝn onto its first k coordinates; i.e., ξx1,…,xn=x1,…,xk.Theorem 6.
LetV⊂ℝk be open and bounded and β0:ClV⟶ℝn−k be a C2 function. We assume that(i)
Z=zα=α,β0α,α∈ClV⊂Ω, and that for each zα∈Z, the solution xt,zα of (34) is T−periodic.(ii)
For eachzα∈Z, there is a fundamental matrix Mzαt of (35) such that the matrix Mzα−10−Mzα−1T has in the upper right corner the k×n−k zero matrix, and in the lower right corner a matrix Δαn−k×n−k with detΔα≠0. We consider the function ℱ:ClV⟶ℝk(36)ℱα=ξ1T0TMzα−1tF1xt,zα,tdt.
If there existsa∈V with ℱa=0 and detdℱ/dαa≠0, then there is a T−periodic solution φt,ε of the system (33) such that φ0,ε⟶za as ε⟶0.
Theorem1 goes back to [10] and [11]; for a shorter proof, see [12].
Note that the periodic orbits provided by Theorem6 are limit cycles.


## 3. Proofs of the Results

### 3.1. Proofs of the Results in Case1: λμδ≠0 and λ≠μ≠δ

#### 3.1.1. Proof of Theorem1

We consider equation (3) and put y=x˙, z=x¨, u=x⃛, and v=x⃜, then equation (3) can be written as(37)x˙=y,y˙=z,z˙=u,u˙=v,v˙=ax+by+cz+du+ev+εFx,y,z,u,v.System (37) with ε=0 has a unique singular point at the origin and the linear part of this system has the eigenvalues ±i, λ, μ, and δ. Using the change of variables(38)XYZUV=λμδ−λμ−λδ−μδλ+μ+δ−100−λμδλμ+λδ+μδ−λ−μ−δ1μδ−μ−δ1+μδ−μ−δ1λδ−λ−δ1+λδ−λ−δ1λμ−λ−μ1+λμ−λ−μ1xyzuv,we transform system (37) into the following system:(39)X˙=−Y,Y˙=X+εF˜X,Y,Z,U,V,Z˙=λZ+εF˜X,Y,Z,U,V,U˙=μU+εF˜X,Y,Z,U,V,V˙=δV+εF˜X,Y,Z,U,V,where(40)F˜=F˜X,Y,Z,U,V=Fa1,a2,a3,a4,a5,a1=λμδ−λ−μ−δXλ2+1μ2+1δ2+1+1−λμ−λδ−μδYλ2+1μ2+1δ2+1+Zλ2+1λ−μλ−δ−Uμ2+1λ−μμ−δ+Vδ2+1λ−δμ−δ,a2=1−λμ−λδ−μδXλ2+1μ2+1δ2+1−λμδ−λ−μ−δYλ2+1μ2+1δ2+1+λZλ2+1λ−μλ−δ−μUμ2+1λ−μμ−δ+δVδ2+1λ−δμ−δ,a3=−λμδ−λ−μ−δXλ2+1μ2+1δ2+1−1−λμ−λδ−μδYλ2+1μ2+1δ2+1+λ2Zλ2+1λ−μλ−δ−μ2Uμ2+1λ−μμ−δ+δ2Vδ2+1λ−δμ−δ,a4=−1−λμ−λδ−μδXλ2+1μ2+1δ2+1+λμδ−λ−μ−δYλ2+1μ2+1δ2+1+λ3Zλ2+1λ−μλ−δ−μ3Uμ2+1λ−μμ−δ+δ3Vδ2+1λ−δμ−δ,a5=λμδ−λ−μ−δXλ2+1μ2+1δ2+1+1−λμ−λδ−μδYλ2+1μ2+1δ2+1+λ4Zλ2+1λ−μλ−δ−μ4Uμ2+1λ−μμ−δ+δ4Vδ2+1λ−δμ−δ.Note that the linear part of system (39) is in the real normal Jordan form of the linear part of system (37). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(41)r˙=εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=λZ+εGr,θ,Z,U,V,U˙=μU+εGr,θ,Z,U,V,V˙=δV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(42)drdθ=εsinθGr,θ,Z,U,V+oε2,dZdθ=λZ+ε1−λZrcosθGr,θ,Z,U,V+oε2,dUdθ=μU+ε1−μUrcosθGr,θ,Z,U,V+oε2,dVdθ=δV+ε1−δVrcosθGr,θ,Z,U,V+oε2.System (42) is now of the same form as system (33) with(43)x=rZUV,t=θ,F0x,θ=0λZμUδV,F1x,θ=sinθGr,θ,Z,U,V1−λZrcosθGr,θ,Z,U,V1−μUrcosθGr,θ,Z,U,V1−δVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (42). System (42) with ε=0 has the 2π−periodic solutions(44)rθZθUθVθ=r0000,∀r0>0.By the notations of Theorem6, we have that k=1 and n=4. Let r1>0 and r2>0; we take V=r1,r2⊂ℝ, α=r0∈r1,r2, and(45)β0:r1,r2⟶ℝ3,:r0↦β0r0=0,0,0.We also take(46)Z=zα=r0,0,0,0,r0∈r1,r2.The fundamental matrixMzαθ of the linear system (42) with ε=0 with respect to the periodic solution zα=r0,0,0,0 satisfying that Mzα0 is the identity matrix is(47)Mzαθ=10000eλθ0000eμθ0000eδθ.We have(48)Mzα−10−Mzα−12π=000001−e−2πλ00001−e−2πμ00001−e−2πδ,which satisfy the assumption (ii) of Theorem 6. Taking(49)ξ:ℝ×ℝ3⟶ℝ,:r,Z,U,V↦ξr,Z,U,V=r,we must compute the function ℱα given by (36), and we obtain(50)ℱα=ℱr0=12π02πsinθGr0,θ,0,0,0dθ=12π02πsinθFA1,A2,A3,A4,A5dθ,and A1,A2,A3,A4, and A5 are given by (6). Then, by Theorem 6, for every simple zero r0∗ of the function ℱr0, there exists a limit cycle r,Z,U,Vθ,ε of system (42) such that(51)r,Z,U,V0,ε⟶r0∗,0,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (41) such that(52)r,θ,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (39) such that(53)X,Y,Z,U,V0,ε⟶r0∗,0,0,0,0whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (7) of equation (8) when ε⟶0.Theorem1 is proved.


#### 3.1.2. Proof of Corollary1

IfFx,x˙,x¨,x⃛,x⃜=x5−x˙4−x¨3−x⃛2−x⃜−1, then we have(54)ℱr0=r0−1+λμ+λδ+μδ161+λ231+μ231+δ23−5r02+41+λ21+μ21+δ2r02+21+λ21+μ21+δ2,which have the real positive simple zero r0∗=25/51+λ21+μ21+δ2 with(55)dfdr0r0∗=−349−1+λμ+λδ+μδ1+λ21+μ21+δ2≠0.The proof of Corollary1 follows directly by applying Theorem 1 and (10) is obtained by substituting r0∗ in (7).


### 3.2. Proofs of the Results in Case2: λ=0,μδ≠0, and μ≠δ

#### 3.2.1. Proof of Theorem2

Ifλ=0, equation (3) can be written as(56)x˙=y,y˙=z,z˙=u,u˙=v,v˙=−μδy+μ+δz−1+μδu+μ+δv+εFx,y,z,u,v.System (56) has a unique singular point at the origin and the eigenvalues of the linear part of this system are ±i, 0, μ, and δ. By the linear transformation(57)XYZUV=0−μδμ+δ−1000μδ−μ−δ1μδ−μ−δ1+μδ−μ−δ10−δ1−δ10−μ1−μ1xyzuv,we transform system (56) into the following system:(58)X˙=−Y,Y˙=X+εF˜X,Y,Z,U,V,Z˙=εF˜X,Y,Z,U,V,U˙=μU+εF˜X,Y,Z,U,V,V˙=δV+εF˜X,Y,Z,U,V,where(59)F˜=F˜X,Y,Z,U,V=Fb1,b2,b3,b4,b5,b1=−μ−δXμ2+1δ2+1+1−μδYμ2+1δ2+1+Zμδ+Uμμ2+1μ−δ−Vδδ2+1μ−δ,b2=1−μδXμ2+1δ2+1−−μ−δYμ2+1δ2+1+Uμ2+1μ−δ−Vδ2+1μ−δ,b3=−−μ−δXμ2+1δ2+1−1−μδYμ2+1δ2+1+μUμ2+1μ−δ−δVδ2+1μ−δ,b4=−1−μδXμ2+1δ2+1+−μ−δYμ2+1δ2+1+μ2Uμ2+1μ−δ−δ2Vδ2+1μ−δ,b5=−μ−δXμ2+1δ2+1+1−μδYμ2+1δ2+1+μ3Uμ2+1μ−δ−δ3Vδ2+1μ−δ.Note that the linear part of system (58) is in the real Jordan normal form of the linear part of system (56). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(60)r˙=εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=εGr,θ,Z,U,V,U˙=μU+εGr,θ,Z,U,V,V˙=δV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(61)drdθ=εsinθGr,θ,Z,U,V+oε2,dZdθ=εGr,θ,Z,U,V+oε2,dUdθ=μU+ε1−μUrcosθGr,θ,Z,U,V+oε2,dVdθ=δV+ε1−δVrcosθGr,θ,Z,U,V+oε2.System (61) is now of the same form as system (33) with(62)x=rZUV,t=θ,F0x,θ=00μUδV,F1x,θ=sinθGr,θ,Z,U,VGr,θ,Z,U,V1−μUrcosθGr,θ,Z,U,V1−δVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (61). System (61) with ε=0 has the 2π−periodic solutions(63)rθZθUθVθ=r0Z000,∀r0>0,∀Z0∈ℝ.By the notations of Theorem6, we have that k=2 and n=4. Let R>0; we take(64)V=r0,Z0:0<r02+Z02<R⊂ℝ2,α=r0,Z0∈V, and(65)β0:ClV⟶ℝ2,:r0,Z0↦β0r0,Z0=0,0.We also take(66)Z=zα=r0,Z0,0,0,r0,Z0∈V.The fundamental matrixMzαθ of the linear system (61) with ε=0 with respect to the periodic solution zα=r0,Z0,0,0 satisfying that Mzα0 is the identity matrix is(67)Mzαθ=1000010000eμθ0000eδθ.We have(68)Mzα−10−Mzα−12π=00000000001−e−2πμ00001−e−2πδ,which satisfy the assumption (ii) of Theorem 6. Taking(69)ξ:ℝ2×ℝ2⟶ℝ2,:r,Z,U,V↦ξr,Z,U,V=r,Z,we must compute the function ℱα given by (36), and we obtain(70)ℱα=ℱr0,Z0=12π02πsinθGr0,θ,Z0,0,0Gr0,θ,Z0,0,0,dθ=ℱ1r0,Z0ℱ2r0,Z0,where(71)ℱ1r0,Z0=12π02πsinθFB1,B2,B3,B4,B5dθ,ℱ2r0,Z0=12π02πFB1,B2,B3,B4,B5dθ,and B1,B2,B3,B4, and B5 are given by (12). Then, by Theorem 6, for every simple zero r0∗,Z0∗ of the function ℱr0,Z0 there exists a limit cycle r,Z,U,Vθ,ε of system (61) such that(72)r,Z,U,V0,ε⟶r0∗,Z0∗,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (60) such that(73)r,θ,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (58) such that(74)X,Y,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (14) of equation (15) when ε⟶0.Theorem2 is proved.


#### 3.2.2. Proof of Corollary2

IfFx,x˙,x¨,x⃛,x⃜=x2−2x+x˙2, then we have(75)ℱ1r0,Z0=r0μδ−1μδ−Z0μδ1+μ21+δ2,ℱ2r0,Z0=r02μ2δ2−Z01+μ21+δ22μδ−Z0μ2δ21+μ21+δ2.The systemℱ1r0,Z0=ℱ2r0,Z0=0 has the zero r0∗,Z0∗=1+μ21+δ2,μδ such that(76)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗=2μδ−1μδ1+μ21+δ2≠0.The proof of Corollary2 follows directly by applying Theorem 2 and (16) is obtained by substituting r0∗,Z0∗ in (14).


### 3.3. Proofs in Case3: λ=0

#### 3.3.1. Proof of Theorem3

Ifλ=0, μ=δ≠0, then equation (3) can be written as(77)x˙=y,y˙=z,z˙=u,u˙=v,v˙=−μ2y+2μz−1+μ2u+2μv+εFx,y,z,u,v.System (77) has a unique singular point at the origin and the eigenvalues of the linear part of this system are ±i, 0, and μ. By the linear transformation(78)XYZUV=0μ2−2μ1000−μ22μ−1μ2−2μ1+μ2−2μ1010100−μ1−μ1xyzuv,we transform system (77) into the following system:(79)X˙=−Y,Y˙=X−εF˜X,Y,Z,U,V,Z˙=εF˜X,Y,Z,U,V,U˙=μU+V,V˙=μV+εF˜X,Y,Z,U,V,where(80)F˜=F˜X,Y,Z,U,V=Fc1,c2,c3,c4,c5,c1=2μXμ2+12+μ2−1Yμ2+12+Zμ2+Uμμ2+1−3μ2+1Vμ2μ2+12,c2=μ2−1Xμ2+12−2μYμ2+12+Uμ2+1−2μVμ2+12,c3=−2μXμ2+12−μ2−1Yμ2+12+μUμ2+1−μ2−1Vμ2+12,c4=−μ2−1Xμ2+12+2μYμ2+12+μ2Uμ2+1+2μVμ2+12,c5=2μXμ2+12+μ2−1Yμ2+12+μ3Uμ2+1+μ2μ2+3Vμ2+12.Note that the linear part of system (81) is in the real Jordan normal form of the linear part of system (77). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(81)r˙=−εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=εGr,θ,Z,U,V,U˙=μU+V,V˙=μV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(82)drdθ=−εsinθGr,θ,Z,U,V+oε2,dZdθ=εGr,θ,Z,U,V+oε2,dUdθ=μU+V+εμU+VrcosθGr,θ,Z,U,V+oε2,dVdθ=μV+ε1+μVrcosθGr,θ,Z,U,V+oε2.System (82) is now of the same form as system (33) with(83)x=rZUV,t=θ,F0x,θ=00μU+VμV,F1x,θ=−sinθGr,θ,Z,U,VGr,θ,Z,U,VμU+VrcosθGr,θ,Z,U,V1+μVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (82). System (82) with ε=0 has the 2π−periodic solutions(84)rθZθUθVθ=r0Z000,∀r0>0,∀Z0∈ℝ.By the notations of Theorem6, we have that k=2 and n=4. Let R>0; we take(85)V=r0,Z0:0<r02+Z02<R⊂ℝ2.α=r0,Z0∈V and(86)β0:ClV⟶ℝ2,:r0,Z0↦β0r0,Z0=0,0.We also take(87)Z=zα=r0,Z0,0,0,r0,Z0∈V.The fundamental matrixMzαθ of the linear system (82) with ε=0 with respect to the periodic solution zα=r0,Z0,0,0 satisfying that Mzα0 is the identity matrix is(88)Mzαθ=1000010000eμθθeμθ000eμθ.We have(89)Mzα−10−Mzα−12π=00000000001−e−2πμ2πe−2πμ0001−e−2πμ,which satisfy the assumption (ii) of Theorem 6. Taking(90)ξ:ℝ2×ℝ2⟶ℝ2,:r,Z,U,V↦ξr,Z,U,V=r,Z,we must compute the function ℱα given by (36), and we obtain(91)ℱα=ℱr0,Z0=12π02π−sinθGr0,θ,Z0,0,0Gr0,θ,Z0,0,0dθ=ℱ1r0,Z0ℱ2r0,Z0,where(92)ℱ1r0,Z0=12π02π−sinθFC1,C2,C3,C4,C5dθ,ℱ2r0,Z0=12π02πFC1,C2,C3,C4,C5dθ,and C1,C2,C3,C4, and C5 are given by (18). Then, by Theorem 6, for every simple zero r0∗,Z0∗ of the function ℱr0,Z0, there exists a limit cycle r,Z,U,Vθ,ε of system (82) such that(93)r,Z,U,V0,ε⟶r0∗,Z0∗,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (81) such that(94)r,θ,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (79) such that(95)X,Y,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (20) of equation (21) when ε⟶0.Theorem3 is proved.


#### 3.3.2. Proof of Corollary3

IfFx,x˙,x¨,x⃛,x⃜=x2−xx˙−1, then we have(96)ℱ1r0,Z0=1−μ−μ2r0Z0μ21+μ22,ℱ2r0,Z0=r0221+μ22+Z02μ4+1.The systemℱ1r0,Z0=ℱ2r0,Z0=0 has the zero r0∗,Z0∗=21+μ2,0 such that(97)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗=2μ2+μ−1μ21+μ22≠0.The proof of Corollary3 follows directly by applying Theorem 3 and (22) is obtained by substituting r0∗,Z0∗ in (20).


### 3.4. Proofs in Case4: λ≠0 and μ=δ≠0

#### 3.4.1. Proof of Theorem4

Ifλ≠0 and μ=δ≠0, then equation (3) can be written as(98)x˙=y,y˙=z,z˙=u,u˙=v,v˙=λμ2x−μ2+2λμy+λ+2μ+λμ2z−μ2+2λμ+1u+λ+2μv+εFx,y,z,u,v.System (98) with ε=0 has a unique singular point at the origin and the linear part of this system has the eigenvalues ±i, λ, and μ. Using the change of variables(99)XYZUV=−λμ2μ2+2λμ−λ−2μ100λμ2−μ2−2λμλ+2μ−1μ2−2μμ2+1−2μ1−λ1−λ10λμ−λ−μ1+λμ−λ−μ1xyzuv,we transform the system (98) into the following system:(100)X˙=−Y,Y˙=X−εF˜X,Y,Z,U,V,Z˙=λZ+εF˜X,Y,Z,U,V,U˙=μU+V,V˙=μV+εF˜X,Y,Z,U,V,where(101)F˜=F˜X,Y,Z,U,V=Fd1,d2,d3,d4,d5,d1=−λμ2+2μ+λXλ2+1μ2+12+μ2+2λμ−1Yλ2+1μ2+12+Zλ2+1λ−μ2−Uμ2+1λ−μ−3μ2−2λμ+1Vμ2+12λ−μ2,d2=μ2+2λμ−1Xλ2+1μ2+12−−λμ2+2μ+λYλ2+1μ2+12+λZλ2+1λ−μ2−μUμ2+1λ−μ−2μ3−λμ2+λVμ2+12λ−μ2,d3=−−λμ2+2μ+λXλ2+1μ2+12−μ2+2λμ−1Yλ2+1μ2+12+λ2Zλ2+1λ−μ2−μ2Uμ2+1λ−μ−μμ3−μ+2λVμ2+12λ−μ2,d4=−μ2+2λμ−1Xλ2+1μ2+12+−λμ2+2μ+λYλ2+1μ2+12+λ3Zλ2+1λ−μ2−μ3Uμ2+1λ−μ−μ2λμ2−2μ+3λVμ2+12λ−μ2,d5=−λμ2+2μ+λXλ2+1μ2+12+μ2+2λμ−1Yλ2+1μ2+12+λ4Zλ2+1λ−μ2−μ4Uμ2+1λ−μ−μ3−μ3+2λμ2−3μ+4λVμ2+12λ−μ2.Note that the linear part of system (100) is in the real normal Jordan form of the linear part of system (98). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(102)r˙=−εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=λZ+εGr,θ,Z,U,V,U˙=μU+V,V˙=μV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(103)drdθ=−εsinθGr,θ,Z,U,V+oε2,dZdθ=λZ+ε1+λZrcosθGr,θ,Z,U,V+oε2,dUdθ=μU+V+εμU+VrcosθGr,θ,Z,U,V+oε2,dVdθ=μV+ε1+μVrcosθGr,θ,Z,U,V+oε2.System (103) is now of the same form as system (33) with(104)x=rZUV,t=θ,F0x,θ=0λZμU+VμV,F1x,θ=−sinθGr,θ,Z,U,V1+λZrcosθGr,θ,Z,U,VμU+VrcosθGr,θ,Z,U,V1+μVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (103). System (103) with ε=0 has the 2π−periodic solutions(105)rθZθUθVθ=r0000,∀r0>0.By the notations of Theorem6, we have that k=1 and n=4. Let r1>0 and r2>0; we take V=r1,r2⊂ℝ, α=r0∈r1,r2, and(106)β0:r1,r2⟶ℝ3,:r0↦β0r0=0,0,0.We also take(107)Z=zα=r0,0,0,0,r0∈r1,r2.The fundamental matrixMzαθ of the linear system (103) with ε=0 with respect to the periodic solution zα=r0,0,0,0 satisfying that Mzα0 is the identity matrix is(108)Mzαθ=10000eλθ0000eμθθeμθ000eμθ.We have(109)Mzα−10−Mzα−12π=000001−e−2πλ00001−e−2πμ2πe−2πμ0001−e−2πμ,which satisfy the assumption (ii) of Theorem 6. Taking(110)ξ:ℝ×ℝ3⟶ℝ,:r,Z,U,V↦ξr,Z,U,V=r,we must compute the function ℱα given by (34), and we obtain(111)ℱα=12π02π−sinθGr0,θ,0,0,0dθ,ℱr0=12π02π−sinθFD1,D2,D3,D4,D5dθ,and D1,D2,D3,D4, and D5 are given by (24). Then, by Theorem 6, for every simple zero r0∗ of the function ℱr0, there exists a limit cycle r,Z,U,Vθ,ε of system (103) such that(112)r,Z,U,V0,ε⟶r0∗,0,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (102) such that(113)r,θ,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system such that(114)X,Y,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (25) of equation (26) when ε⟶0.Theorem4 is proved.


#### 3.4.2. Proof of Corollary4

IfFx,x˙,x¨,x⃛,x⃜=x3−x−1, then we have(115)ℱr0=r0−1+2λμ+μ241+λ21+μ22−3r0281+λ221+μ24,which have the real positive simple zero r0∗=23/31+λ21+μ2 with(116)dfdr0r0∗=1−2λμ−μ21+λ21+μ22≠0.The proof of Corollary4 follows directly by applying Theorem 4 and (27) is obtained by substituting r0∗ in (25).


### 3.5. Proofs in Case5: λ=μ=δ≠0

#### 3.5.1. Proof of Theorem5

Ifλ=μ=δ≠0, then equation (3) can be written as(117)x˙=y,y˙=z,z˙=u,u˙=v,v˙=λ3x−3λ2y+3λ+λ3z−3λ2+1u+3λv+εFx,y,z,u,v.System (117) with ε=0 has a unique singular point at the origin and the linear part of this system has the eigenvalues ±i and λ. Using the change of variables(118)XYZUV=λ3−3λ23λ−100−λ33λ2−3λ110100−λ1−λ10λ2−2λλ2+1−2λ1xyzuv,we transform system (117) into the following system:(119)X˙=−Y,Y˙=X+εF˜X,Y,Z,U,V,Z˙=λZ+U,U˙=λU+V,V˙=λV+εF˜X,Y,Z,U,V,where(120)F˜=F˜X,Y,Z,U,V=Fe1,e2,e3,e4,e5,e1=λλ2−3X+1−3λ2Y+λ2+12Z−2λλ2+1U+3λ2−1Vλ2+13,e2=1−3λ2X−λλ2−3Y+λλ2+12Z+1−λ4U+λλ2−3Vλ2+13,e3=−λλ2−3X−1−3λ2Y+λ2λ2+12Z+2λλ2+1U+1−3λ2Vλ2+13,e4=−1−3λ2X+λλ2−3Y+λ3λ2+12Z+λ2λ2+3λ2+1U−λλ2−3Vλ2+13,e5=λλ2−3X+1−3λ2Y+λ4λ2+12Z+2λ3λ2+2λ2+1U+λ23λ2+λ4+6Vλ2+13.Note that the linear part of system (119) is in the real normal Jordan form of the linear part of system (51). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(121)r˙=εsinθGr,θ,Z,U,V,θ.=1+εrcosθGr,θ,Z,U,V,Z˙=λZ+U,U˙=λU+V,V˙=λV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(122)drdθ=εsinθGr,θ,Z,U,V+oε2,dZdθ=λZ+U−ελZ+UrcosθGr,θ,Z,U,V+oε2,dUdθ=λU+V−ελU+VrcosθGr,θ,Z,U,V+oε2,dVdθ=λV+ε1−λVrcosθGr,θ,Z,U,V+oε2.System (122) is now of the same form as system (33) with(123)x=rZUV,t=θ,F0x,θ=0λZ+UλU+VλV,F1x,θ=−sinθGr,θ,Z,U,V−λZ+UrcosθGr,θ,Z,U,V−λU+VrcosθGr,θ,Z,U,V1−λVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (122). System (122) with ε=0 has the 2π−periodic solutions(124)rθZθUθVθ=r0000,∀r0>0.By the notations of Theorem6, we have that k=1 and n=4. Let r1>0 and r2>0; we take V=r1,r2⊂ℝ, α=r0∈r1,r2, and(125)β0:r1,r2⟶ℝ3,:r0↦β0r0=0,0,0.We also take(126)Z=zα=r0,0,0,0,r0∈r1,r2.The fundamental matrixMzαθ of the linear system (122) with ε=0 with respect to the periodic solution zα=r0,0,0,0 satisfying that Mzα0 is the identity matrix is(127)Mzαθ=10000eλθθeλθθ22eλθ00eλθθeλθ000eλθ.We have(128)Mzα−10−Mzα−12π=000001−e−2πλ2πe−2πλ−2π2e−2πλ001−e−2πλ2πe−2πλ0001−e−2πλ,which satisfy the assumption (ii) of Theorem 6. Taking(129)ξ:ℝ×ℝ3⟶ℝ,:r,Z,U,V↦ξr,Z,U,V=r,we must compute the function ℱα given by (36), and we obtain(130)ℱr0=12π02πsinθGr0,θ,0,0,0dθ,ℱα=12π02πsinθFE1,E2,E3,E4,E5dθ,and E1,E2,E3,E4, and E5 are given by (29). Then, by Theorem 6, for every simple zero r0∗ of the function ℱr0, there exists a limit cycle r,Z,U,Vθ,ε of system (122) such that(131)r,Z,U,V0,ε⟶r0∗,0,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (121) such that(132)r,θ,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (119) such that(133)X,Y,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (30) of equation (31) when ε⟶0.Theorem5 is proved.


#### 3.5.2. Proof of Corollary5

IfFx,x˙,x¨,x⃛,x⃜=−x¨2x⃜2+x3+x2−x˙x¨+x⃛−1, then we have(134)ℱr0=r03r023λ2−1−4λλ2−31+λ2381+λ26,which have the real positive simple zero r0∗=23/3λ3λ2−1λ2−31+λ21+λ2/3λ2−1 with(135)dfdr0r0∗=λλ2−31+λ23≠0.The proof of Corollary5 follows directly by applying Theorem 5 and (32) is obtained by substituting r0∗ in (30).


## 3.1. Proofs of the Results in Case1: λμδ≠0 and λ≠μ≠δ

### 3.1.1. Proof of Theorem1

We consider equation (3) and put y=x˙, z=x¨, u=x⃛, and v=x⃜, then equation (3) can be written as(37)x˙=y,y˙=z,z˙=u,u˙=v,v˙=ax+by+cz+du+ev+εFx,y,z,u,v.System (37) with ε=0 has a unique singular point at the origin and the linear part of this system has the eigenvalues ±i, λ, μ, and δ. Using the change of variables(38)XYZUV=λμδ−λμ−λδ−μδλ+μ+δ−100−λμδλμ+λδ+μδ−λ−μ−δ1μδ−μ−δ1+μδ−μ−δ1λδ−λ−δ1+λδ−λ−δ1λμ−λ−μ1+λμ−λ−μ1xyzuv,we transform system (37) into the following system:(39)X˙=−Y,Y˙=X+εF˜X,Y,Z,U,V,Z˙=λZ+εF˜X,Y,Z,U,V,U˙=μU+εF˜X,Y,Z,U,V,V˙=δV+εF˜X,Y,Z,U,V,where(40)F˜=F˜X,Y,Z,U,V=Fa1,a2,a3,a4,a5,a1=λμδ−λ−μ−δXλ2+1μ2+1δ2+1+1−λμ−λδ−μδYλ2+1μ2+1δ2+1+Zλ2+1λ−μλ−δ−Uμ2+1λ−μμ−δ+Vδ2+1λ−δμ−δ,a2=1−λμ−λδ−μδXλ2+1μ2+1δ2+1−λμδ−λ−μ−δYλ2+1μ2+1δ2+1+λZλ2+1λ−μλ−δ−μUμ2+1λ−μμ−δ+δVδ2+1λ−δμ−δ,a3=−λμδ−λ−μ−δXλ2+1μ2+1δ2+1−1−λμ−λδ−μδYλ2+1μ2+1δ2+1+λ2Zλ2+1λ−μλ−δ−μ2Uμ2+1λ−μμ−δ+δ2Vδ2+1λ−δμ−δ,a4=−1−λμ−λδ−μδXλ2+1μ2+1δ2+1+λμδ−λ−μ−δYλ2+1μ2+1δ2+1+λ3Zλ2+1λ−μλ−δ−μ3Uμ2+1λ−μμ−δ+δ3Vδ2+1λ−δμ−δ,a5=λμδ−λ−μ−δXλ2+1μ2+1δ2+1+1−λμ−λδ−μδYλ2+1μ2+1δ2+1+λ4Zλ2+1λ−μλ−δ−μ4Uμ2+1λ−μμ−δ+δ4Vδ2+1λ−δμ−δ.Note that the linear part of system (39) is in the real normal Jordan form of the linear part of system (37). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(41)r˙=εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=λZ+εGr,θ,Z,U,V,U˙=μU+εGr,θ,Z,U,V,V˙=δV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(42)drdθ=εsinθGr,θ,Z,U,V+oε2,dZdθ=λZ+ε1−λZrcosθGr,θ,Z,U,V+oε2,dUdθ=μU+ε1−μUrcosθGr,θ,Z,U,V+oε2,dVdθ=δV+ε1−δVrcosθGr,θ,Z,U,V+oε2.System (42) is now of the same form as system (33) with(43)x=rZUV,t=θ,F0x,θ=0λZμUδV,F1x,θ=sinθGr,θ,Z,U,V1−λZrcosθGr,θ,Z,U,V1−μUrcosθGr,θ,Z,U,V1−δVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (42). System (42) with ε=0 has the 2π−periodic solutions(44)rθZθUθVθ=r0000,∀r0>0.By the notations of Theorem6, we have that k=1 and n=4. Let r1>0 and r2>0; we take V=r1,r2⊂ℝ, α=r0∈r1,r2, and(45)β0:r1,r2⟶ℝ3,:r0↦β0r0=0,0,0.We also take(46)Z=zα=r0,0,0,0,r0∈r1,r2.The fundamental matrixMzαθ of the linear system (42) with ε=0 with respect to the periodic solution zα=r0,0,0,0 satisfying that Mzα0 is the identity matrix is(47)Mzαθ=10000eλθ0000eμθ0000eδθ.We have(48)Mzα−10−Mzα−12π=000001−e−2πλ00001−e−2πμ00001−e−2πδ,which satisfy the assumption (ii) of Theorem 6. Taking(49)ξ:ℝ×ℝ3⟶ℝ,:r,Z,U,V↦ξr,Z,U,V=r,we must compute the function ℱα given by (36), and we obtain(50)ℱα=ℱr0=12π02πsinθGr0,θ,0,0,0dθ=12π02πsinθFA1,A2,A3,A4,A5dθ,and A1,A2,A3,A4, and A5 are given by (6). Then, by Theorem 6, for every simple zero r0∗ of the function ℱr0, there exists a limit cycle r,Z,U,Vθ,ε of system (42) such that(51)r,Z,U,V0,ε⟶r0∗,0,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (41) such that(52)r,θ,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (39) such that(53)X,Y,Z,U,V0,ε⟶r0∗,0,0,0,0whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (7) of equation (8) when ε⟶0.Theorem1 is proved.


### 3.1.2. Proof of Corollary1

IfFx,x˙,x¨,x⃛,x⃜=x5−x˙4−x¨3−x⃛2−x⃜−1, then we have(54)ℱr0=r0−1+λμ+λδ+μδ161+λ231+μ231+δ23−5r02+41+λ21+μ21+δ2r02+21+λ21+μ21+δ2,which have the real positive simple zero r0∗=25/51+λ21+μ21+δ2 with(55)dfdr0r0∗=−349−1+λμ+λδ+μδ1+λ21+μ21+δ2≠0.The proof of Corollary1 follows directly by applying Theorem 1 and (10) is obtained by substituting r0∗ in (7).


## 3.1.1. Proof of Theorem1

We consider equation (3) and put y=x˙, z=x¨, u=x⃛, and v=x⃜, then equation (3) can be written as(37)x˙=y,y˙=z,z˙=u,u˙=v,v˙=ax+by+cz+du+ev+εFx,y,z,u,v.System (37) with ε=0 has a unique singular point at the origin and the linear part of this system has the eigenvalues ±i, λ, μ, and δ. Using the change of variables(38)XYZUV=λμδ−λμ−λδ−μδλ+μ+δ−100−λμδλμ+λδ+μδ−λ−μ−δ1μδ−μ−δ1+μδ−μ−δ1λδ−λ−δ1+λδ−λ−δ1λμ−λ−μ1+λμ−λ−μ1xyzuv,we transform system (37) into the following system:(39)X˙=−Y,Y˙=X+εF˜X,Y,Z,U,V,Z˙=λZ+εF˜X,Y,Z,U,V,U˙=μU+εF˜X,Y,Z,U,V,V˙=δV+εF˜X,Y,Z,U,V,where(40)F˜=F˜X,Y,Z,U,V=Fa1,a2,a3,a4,a5,a1=λμδ−λ−μ−δXλ2+1μ2+1δ2+1+1−λμ−λδ−μδYλ2+1μ2+1δ2+1+Zλ2+1λ−μλ−δ−Uμ2+1λ−μμ−δ+Vδ2+1λ−δμ−δ,a2=1−λμ−λδ−μδXλ2+1μ2+1δ2+1−λμδ−λ−μ−δYλ2+1μ2+1δ2+1+λZλ2+1λ−μλ−δ−μUμ2+1λ−μμ−δ+δVδ2+1λ−δμ−δ,a3=−λμδ−λ−μ−δXλ2+1μ2+1δ2+1−1−λμ−λδ−μδYλ2+1μ2+1δ2+1+λ2Zλ2+1λ−μλ−δ−μ2Uμ2+1λ−μμ−δ+δ2Vδ2+1λ−δμ−δ,a4=−1−λμ−λδ−μδXλ2+1μ2+1δ2+1+λμδ−λ−μ−δYλ2+1μ2+1δ2+1+λ3Zλ2+1λ−μλ−δ−μ3Uμ2+1λ−μμ−δ+δ3Vδ2+1λ−δμ−δ,a5=λμδ−λ−μ−δXλ2+1μ2+1δ2+1+1−λμ−λδ−μδYλ2+1μ2+1δ2+1+λ4Zλ2+1λ−μλ−δ−μ4Uμ2+1λ−μμ−δ+δ4Vδ2+1λ−δμ−δ.Note that the linear part of system (39) is in the real normal Jordan form of the linear part of system (37). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(41)r˙=εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=λZ+εGr,θ,Z,U,V,U˙=μU+εGr,θ,Z,U,V,V˙=δV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(42)drdθ=εsinθGr,θ,Z,U,V+oε2,dZdθ=λZ+ε1−λZrcosθGr,θ,Z,U,V+oε2,dUdθ=μU+ε1−μUrcosθGr,θ,Z,U,V+oε2,dVdθ=δV+ε1−δVrcosθGr,θ,Z,U,V+oε2.System (42) is now of the same form as system (33) with(43)x=rZUV,t=θ,F0x,θ=0λZμUδV,F1x,θ=sinθGr,θ,Z,U,V1−λZrcosθGr,θ,Z,U,V1−μUrcosθGr,θ,Z,U,V1−δVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (42). System (42) with ε=0 has the 2π−periodic solutions(44)rθZθUθVθ=r0000,∀r0>0.By the notations of Theorem6, we have that k=1 and n=4. Let r1>0 and r2>0; we take V=r1,r2⊂ℝ, α=r0∈r1,r2, and(45)β0:r1,r2⟶ℝ3,:r0↦β0r0=0,0,0.We also take(46)Z=zα=r0,0,0,0,r0∈r1,r2.The fundamental matrixMzαθ of the linear system (42) with ε=0 with respect to the periodic solution zα=r0,0,0,0 satisfying that Mzα0 is the identity matrix is(47)Mzαθ=10000eλθ0000eμθ0000eδθ.We have(48)Mzα−10−Mzα−12π=000001−e−2πλ00001−e−2πμ00001−e−2πδ,which satisfy the assumption (ii) of Theorem 6. Taking(49)ξ:ℝ×ℝ3⟶ℝ,:r,Z,U,V↦ξr,Z,U,V=r,we must compute the function ℱα given by (36), and we obtain(50)ℱα=ℱr0=12π02πsinθGr0,θ,0,0,0dθ=12π02πsinθFA1,A2,A3,A4,A5dθ,and A1,A2,A3,A4, and A5 are given by (6). Then, by Theorem 6, for every simple zero r0∗ of the function ℱr0, there exists a limit cycle r,Z,U,Vθ,ε of system (42) such that(51)r,Z,U,V0,ε⟶r0∗,0,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (41) such that(52)r,θ,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (39) such that(53)X,Y,Z,U,V0,ε⟶r0∗,0,0,0,0whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (7) of equation (8) when ε⟶0.Theorem1 is proved.


## 3.1.2. Proof of Corollary1

IfFx,x˙,x¨,x⃛,x⃜=x5−x˙4−x¨3−x⃛2−x⃜−1, then we have(54)ℱr0=r0−1+λμ+λδ+μδ161+λ231+μ231+δ23−5r02+41+λ21+μ21+δ2r02+21+λ21+μ21+δ2,which have the real positive simple zero r0∗=25/51+λ21+μ21+δ2 with(55)dfdr0r0∗=−349−1+λμ+λδ+μδ1+λ21+μ21+δ2≠0.The proof of Corollary1 follows directly by applying Theorem 1 and (10) is obtained by substituting r0∗ in (7).


## 3.2. Proofs of the Results in Case2: λ=0,μδ≠0, and μ≠δ

### 3.2.1. Proof of Theorem2

Ifλ=0, equation (3) can be written as(56)x˙=y,y˙=z,z˙=u,u˙=v,v˙=−μδy+μ+δz−1+μδu+μ+δv+εFx,y,z,u,v.System (56) has a unique singular point at the origin and the eigenvalues of the linear part of this system are ±i, 0, μ, and δ. By the linear transformation(57)XYZUV=0−μδμ+δ−1000μδ−μ−δ1μδ−μ−δ1+μδ−μ−δ10−δ1−δ10−μ1−μ1xyzuv,we transform system (56) into the following system:(58)X˙=−Y,Y˙=X+εF˜X,Y,Z,U,V,Z˙=εF˜X,Y,Z,U,V,U˙=μU+εF˜X,Y,Z,U,V,V˙=δV+εF˜X,Y,Z,U,V,where(59)F˜=F˜X,Y,Z,U,V=Fb1,b2,b3,b4,b5,b1=−μ−δXμ2+1δ2+1+1−μδYμ2+1δ2+1+Zμδ+Uμμ2+1μ−δ−Vδδ2+1μ−δ,b2=1−μδXμ2+1δ2+1−−μ−δYμ2+1δ2+1+Uμ2+1μ−δ−Vδ2+1μ−δ,b3=−−μ−δXμ2+1δ2+1−1−μδYμ2+1δ2+1+μUμ2+1μ−δ−δVδ2+1μ−δ,b4=−1−μδXμ2+1δ2+1+−μ−δYμ2+1δ2+1+μ2Uμ2+1μ−δ−δ2Vδ2+1μ−δ,b5=−μ−δXμ2+1δ2+1+1−μδYμ2+1δ2+1+μ3Uμ2+1μ−δ−δ3Vδ2+1μ−δ.Note that the linear part of system (58) is in the real Jordan normal form of the linear part of system (56). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(60)r˙=εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=εGr,θ,Z,U,V,U˙=μU+εGr,θ,Z,U,V,V˙=δV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(61)drdθ=εsinθGr,θ,Z,U,V+oε2,dZdθ=εGr,θ,Z,U,V+oε2,dUdθ=μU+ε1−μUrcosθGr,θ,Z,U,V+oε2,dVdθ=δV+ε1−δVrcosθGr,θ,Z,U,V+oε2.System (61) is now of the same form as system (33) with(62)x=rZUV,t=θ,F0x,θ=00μUδV,F1x,θ=sinθGr,θ,Z,U,VGr,θ,Z,U,V1−μUrcosθGr,θ,Z,U,V1−δVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (61). System (61) with ε=0 has the 2π−periodic solutions(63)rθZθUθVθ=r0Z000,∀r0>0,∀Z0∈ℝ.By the notations of Theorem6, we have that k=2 and n=4. Let R>0; we take(64)V=r0,Z0:0<r02+Z02<R⊂ℝ2,α=r0,Z0∈V, and(65)β0:ClV⟶ℝ2,:r0,Z0↦β0r0,Z0=0,0.We also take(66)Z=zα=r0,Z0,0,0,r0,Z0∈V.The fundamental matrixMzαθ of the linear system (61) with ε=0 with respect to the periodic solution zα=r0,Z0,0,0 satisfying that Mzα0 is the identity matrix is(67)Mzαθ=1000010000eμθ0000eδθ.We have(68)Mzα−10−Mzα−12π=00000000001−e−2πμ00001−e−2πδ,which satisfy the assumption (ii) of Theorem 6. Taking(69)ξ:ℝ2×ℝ2⟶ℝ2,:r,Z,U,V↦ξr,Z,U,V=r,Z,we must compute the function ℱα given by (36), and we obtain(70)ℱα=ℱr0,Z0=12π02πsinθGr0,θ,Z0,0,0Gr0,θ,Z0,0,0,dθ=ℱ1r0,Z0ℱ2r0,Z0,where(71)ℱ1r0,Z0=12π02πsinθFB1,B2,B3,B4,B5dθ,ℱ2r0,Z0=12π02πFB1,B2,B3,B4,B5dθ,and B1,B2,B3,B4, and B5 are given by (12). Then, by Theorem 6, for every simple zero r0∗,Z0∗ of the function ℱr0,Z0 there exists a limit cycle r,Z,U,Vθ,ε of system (61) such that(72)r,Z,U,V0,ε⟶r0∗,Z0∗,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (60) such that(73)r,θ,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (58) such that(74)X,Y,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (14) of equation (15) when ε⟶0.Theorem2 is proved.


### 3.2.2. Proof of Corollary2

IfFx,x˙,x¨,x⃛,x⃜=x2−2x+x˙2, then we have(75)ℱ1r0,Z0=r0μδ−1μδ−Z0μδ1+μ21+δ2,ℱ2r0,Z0=r02μ2δ2−Z01+μ21+δ22μδ−Z0μ2δ21+μ21+δ2.The systemℱ1r0,Z0=ℱ2r0,Z0=0 has the zero r0∗,Z0∗=1+μ21+δ2,μδ such that(76)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗=2μδ−1μδ1+μ21+δ2≠0.The proof of Corollary2 follows directly by applying Theorem 2 and (16) is obtained by substituting r0∗,Z0∗ in (14).


## 3.2.1. Proof of Theorem2

Ifλ=0, equation (3) can be written as(56)x˙=y,y˙=z,z˙=u,u˙=v,v˙=−μδy+μ+δz−1+μδu+μ+δv+εFx,y,z,u,v.System (56) has a unique singular point at the origin and the eigenvalues of the linear part of this system are ±i, 0, μ, and δ. By the linear transformation(57)XYZUV=0−μδμ+δ−1000μδ−μ−δ1μδ−μ−δ1+μδ−μ−δ10−δ1−δ10−μ1−μ1xyzuv,we transform system (56) into the following system:(58)X˙=−Y,Y˙=X+εF˜X,Y,Z,U,V,Z˙=εF˜X,Y,Z,U,V,U˙=μU+εF˜X,Y,Z,U,V,V˙=δV+εF˜X,Y,Z,U,V,where(59)F˜=F˜X,Y,Z,U,V=Fb1,b2,b3,b4,b5,b1=−μ−δXμ2+1δ2+1+1−μδYμ2+1δ2+1+Zμδ+Uμμ2+1μ−δ−Vδδ2+1μ−δ,b2=1−μδXμ2+1δ2+1−−μ−δYμ2+1δ2+1+Uμ2+1μ−δ−Vδ2+1μ−δ,b3=−−μ−δXμ2+1δ2+1−1−μδYμ2+1δ2+1+μUμ2+1μ−δ−δVδ2+1μ−δ,b4=−1−μδXμ2+1δ2+1+−μ−δYμ2+1δ2+1+μ2Uμ2+1μ−δ−δ2Vδ2+1μ−δ,b5=−μ−δXμ2+1δ2+1+1−μδYμ2+1δ2+1+μ3Uμ2+1μ−δ−δ3Vδ2+1μ−δ.Note that the linear part of system (58) is in the real Jordan normal form of the linear part of system (56). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(60)r˙=εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=εGr,θ,Z,U,V,U˙=μU+εGr,θ,Z,U,V,V˙=δV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(61)drdθ=εsinθGr,θ,Z,U,V+oε2,dZdθ=εGr,θ,Z,U,V+oε2,dUdθ=μU+ε1−μUrcosθGr,θ,Z,U,V+oε2,dVdθ=δV+ε1−δVrcosθGr,θ,Z,U,V+oε2.System (61) is now of the same form as system (33) with(62)x=rZUV,t=θ,F0x,θ=00μUδV,F1x,θ=sinθGr,θ,Z,U,VGr,θ,Z,U,V1−μUrcosθGr,θ,Z,U,V1−δVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (61). System (61) with ε=0 has the 2π−periodic solutions(63)rθZθUθVθ=r0Z000,∀r0>0,∀Z0∈ℝ.By the notations of Theorem6, we have that k=2 and n=4. Let R>0; we take(64)V=r0,Z0:0<r02+Z02<R⊂ℝ2,α=r0,Z0∈V, and(65)β0:ClV⟶ℝ2,:r0,Z0↦β0r0,Z0=0,0.We also take(66)Z=zα=r0,Z0,0,0,r0,Z0∈V.The fundamental matrixMzαθ of the linear system (61) with ε=0 with respect to the periodic solution zα=r0,Z0,0,0 satisfying that Mzα0 is the identity matrix is(67)Mzαθ=1000010000eμθ0000eδθ.We have(68)Mzα−10−Mzα−12π=00000000001−e−2πμ00001−e−2πδ,which satisfy the assumption (ii) of Theorem 6. Taking(69)ξ:ℝ2×ℝ2⟶ℝ2,:r,Z,U,V↦ξr,Z,U,V=r,Z,we must compute the function ℱα given by (36), and we obtain(70)ℱα=ℱr0,Z0=12π02πsinθGr0,θ,Z0,0,0Gr0,θ,Z0,0,0,dθ=ℱ1r0,Z0ℱ2r0,Z0,where(71)ℱ1r0,Z0=12π02πsinθFB1,B2,B3,B4,B5dθ,ℱ2r0,Z0=12π02πFB1,B2,B3,B4,B5dθ,and B1,B2,B3,B4, and B5 are given by (12). Then, by Theorem 6, for every simple zero r0∗,Z0∗ of the function ℱr0,Z0 there exists a limit cycle r,Z,U,Vθ,ε of system (61) such that(72)r,Z,U,V0,ε⟶r0∗,Z0∗,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (60) such that(73)r,θ,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (58) such that(74)X,Y,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (14) of equation (15) when ε⟶0.Theorem2 is proved.


## 3.2.2. Proof of Corollary2

IfFx,x˙,x¨,x⃛,x⃜=x2−2x+x˙2, then we have(75)ℱ1r0,Z0=r0μδ−1μδ−Z0μδ1+μ21+δ2,ℱ2r0,Z0=r02μ2δ2−Z01+μ21+δ22μδ−Z0μ2δ21+μ21+δ2.The systemℱ1r0,Z0=ℱ2r0,Z0=0 has the zero r0∗,Z0∗=1+μ21+δ2,μδ such that(76)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗=2μδ−1μδ1+μ21+δ2≠0.The proof of Corollary2 follows directly by applying Theorem 2 and (16) is obtained by substituting r0∗,Z0∗ in (14).


## 3.3. Proofs in Case3: λ=0

### 3.3.1. Proof of Theorem3

Ifλ=0, μ=δ≠0, then equation (3) can be written as(77)x˙=y,y˙=z,z˙=u,u˙=v,v˙=−μ2y+2μz−1+μ2u+2μv+εFx,y,z,u,v.System (77) has a unique singular point at the origin and the eigenvalues of the linear part of this system are ±i, 0, and μ. By the linear transformation(78)XYZUV=0μ2−2μ1000−μ22μ−1μ2−2μ1+μ2−2μ1010100−μ1−μ1xyzuv,we transform system (77) into the following system:(79)X˙=−Y,Y˙=X−εF˜X,Y,Z,U,V,Z˙=εF˜X,Y,Z,U,V,U˙=μU+V,V˙=μV+εF˜X,Y,Z,U,V,where(80)F˜=F˜X,Y,Z,U,V=Fc1,c2,c3,c4,c5,c1=2μXμ2+12+μ2−1Yμ2+12+Zμ2+Uμμ2+1−3μ2+1Vμ2μ2+12,c2=μ2−1Xμ2+12−2μYμ2+12+Uμ2+1−2μVμ2+12,c3=−2μXμ2+12−μ2−1Yμ2+12+μUμ2+1−μ2−1Vμ2+12,c4=−μ2−1Xμ2+12+2μYμ2+12+μ2Uμ2+1+2μVμ2+12,c5=2μXμ2+12+μ2−1Yμ2+12+μ3Uμ2+1+μ2μ2+3Vμ2+12.Note that the linear part of system (81) is in the real Jordan normal form of the linear part of system (77). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(81)r˙=−εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=εGr,θ,Z,U,V,U˙=μU+V,V˙=μV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(82)drdθ=−εsinθGr,θ,Z,U,V+oε2,dZdθ=εGr,θ,Z,U,V+oε2,dUdθ=μU+V+εμU+VrcosθGr,θ,Z,U,V+oε2,dVdθ=μV+ε1+μVrcosθGr,θ,Z,U,V+oε2.System (82) is now of the same form as system (33) with(83)x=rZUV,t=θ,F0x,θ=00μU+VμV,F1x,θ=−sinθGr,θ,Z,U,VGr,θ,Z,U,VμU+VrcosθGr,θ,Z,U,V1+μVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (82). System (82) with ε=0 has the 2π−periodic solutions(84)rθZθUθVθ=r0Z000,∀r0>0,∀Z0∈ℝ.By the notations of Theorem6, we have that k=2 and n=4. Let R>0; we take(85)V=r0,Z0:0<r02+Z02<R⊂ℝ2.α=r0,Z0∈V and(86)β0:ClV⟶ℝ2,:r0,Z0↦β0r0,Z0=0,0.We also take(87)Z=zα=r0,Z0,0,0,r0,Z0∈V.The fundamental matrixMzαθ of the linear system (82) with ε=0 with respect to the periodic solution zα=r0,Z0,0,0 satisfying that Mzα0 is the identity matrix is(88)Mzαθ=1000010000eμθθeμθ000eμθ.We have(89)Mzα−10−Mzα−12π=00000000001−e−2πμ2πe−2πμ0001−e−2πμ,which satisfy the assumption (ii) of Theorem 6. Taking(90)ξ:ℝ2×ℝ2⟶ℝ2,:r,Z,U,V↦ξr,Z,U,V=r,Z,we must compute the function ℱα given by (36), and we obtain(91)ℱα=ℱr0,Z0=12π02π−sinθGr0,θ,Z0,0,0Gr0,θ,Z0,0,0dθ=ℱ1r0,Z0ℱ2r0,Z0,where(92)ℱ1r0,Z0=12π02π−sinθFC1,C2,C3,C4,C5dθ,ℱ2r0,Z0=12π02πFC1,C2,C3,C4,C5dθ,and C1,C2,C3,C4, and C5 are given by (18). Then, by Theorem 6, for every simple zero r0∗,Z0∗ of the function ℱr0,Z0, there exists a limit cycle r,Z,U,Vθ,ε of system (82) such that(93)r,Z,U,V0,ε⟶r0∗,Z0∗,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (81) such that(94)r,θ,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (79) such that(95)X,Y,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (20) of equation (21) when ε⟶0.Theorem3 is proved.


### 3.3.2. Proof of Corollary3

IfFx,x˙,x¨,x⃛,x⃜=x2−xx˙−1, then we have(96)ℱ1r0,Z0=1−μ−μ2r0Z0μ21+μ22,ℱ2r0,Z0=r0221+μ22+Z02μ4+1.The systemℱ1r0,Z0=ℱ2r0,Z0=0 has the zero r0∗,Z0∗=21+μ2,0 such that(97)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗=2μ2+μ−1μ21+μ22≠0.The proof of Corollary3 follows directly by applying Theorem 3 and (22) is obtained by substituting r0∗,Z0∗ in (20).


## 3.3.1. Proof of Theorem3

Ifλ=0, μ=δ≠0, then equation (3) can be written as(77)x˙=y,y˙=z,z˙=u,u˙=v,v˙=−μ2y+2μz−1+μ2u+2μv+εFx,y,z,u,v.System (77) has a unique singular point at the origin and the eigenvalues of the linear part of this system are ±i, 0, and μ. By the linear transformation(78)XYZUV=0μ2−2μ1000−μ22μ−1μ2−2μ1+μ2−2μ1010100−μ1−μ1xyzuv,we transform system (77) into the following system:(79)X˙=−Y,Y˙=X−εF˜X,Y,Z,U,V,Z˙=εF˜X,Y,Z,U,V,U˙=μU+V,V˙=μV+εF˜X,Y,Z,U,V,where(80)F˜=F˜X,Y,Z,U,V=Fc1,c2,c3,c4,c5,c1=2μXμ2+12+μ2−1Yμ2+12+Zμ2+Uμμ2+1−3μ2+1Vμ2μ2+12,c2=μ2−1Xμ2+12−2μYμ2+12+Uμ2+1−2μVμ2+12,c3=−2μXμ2+12−μ2−1Yμ2+12+μUμ2+1−μ2−1Vμ2+12,c4=−μ2−1Xμ2+12+2μYμ2+12+μ2Uμ2+1+2μVμ2+12,c5=2μXμ2+12+μ2−1Yμ2+12+μ3Uμ2+1+μ2μ2+3Vμ2+12.Note that the linear part of system (81) is in the real Jordan normal form of the linear part of system (77). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(81)r˙=−εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=εGr,θ,Z,U,V,U˙=μU+V,V˙=μV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(82)drdθ=−εsinθGr,θ,Z,U,V+oε2,dZdθ=εGr,θ,Z,U,V+oε2,dUdθ=μU+V+εμU+VrcosθGr,θ,Z,U,V+oε2,dVdθ=μV+ε1+μVrcosθGr,θ,Z,U,V+oε2.System (82) is now of the same form as system (33) with(83)x=rZUV,t=θ,F0x,θ=00μU+VμV,F1x,θ=−sinθGr,θ,Z,U,VGr,θ,Z,U,VμU+VrcosθGr,θ,Z,U,V1+μVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (82). System (82) with ε=0 has the 2π−periodic solutions(84)rθZθUθVθ=r0Z000,∀r0>0,∀Z0∈ℝ.By the notations of Theorem6, we have that k=2 and n=4. Let R>0; we take(85)V=r0,Z0:0<r02+Z02<R⊂ℝ2.α=r0,Z0∈V and(86)β0:ClV⟶ℝ2,:r0,Z0↦β0r0,Z0=0,0.We also take(87)Z=zα=r0,Z0,0,0,r0,Z0∈V.The fundamental matrixMzαθ of the linear system (82) with ε=0 with respect to the periodic solution zα=r0,Z0,0,0 satisfying that Mzα0 is the identity matrix is(88)Mzαθ=1000010000eμθθeμθ000eμθ.We have(89)Mzα−10−Mzα−12π=00000000001−e−2πμ2πe−2πμ0001−e−2πμ,which satisfy the assumption (ii) of Theorem 6. Taking(90)ξ:ℝ2×ℝ2⟶ℝ2,:r,Z,U,V↦ξr,Z,U,V=r,Z,we must compute the function ℱα given by (36), and we obtain(91)ℱα=ℱr0,Z0=12π02π−sinθGr0,θ,Z0,0,0Gr0,θ,Z0,0,0dθ=ℱ1r0,Z0ℱ2r0,Z0,where(92)ℱ1r0,Z0=12π02π−sinθFC1,C2,C3,C4,C5dθ,ℱ2r0,Z0=12π02πFC1,C2,C3,C4,C5dθ,and C1,C2,C3,C4, and C5 are given by (18). Then, by Theorem 6, for every simple zero r0∗,Z0∗ of the function ℱr0,Z0, there exists a limit cycle r,Z,U,Vθ,ε of system (82) such that(93)r,Z,U,V0,ε⟶r0∗,Z0∗,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (81) such that(94)r,θ,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (79) such that(95)X,Y,Z,U,V0,ε⟶r0∗,Z0∗,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (20) of equation (21) when ε⟶0.Theorem3 is proved.


## 3.3.2. Proof of Corollary3

IfFx,x˙,x¨,x⃛,x⃜=x2−xx˙−1, then we have(96)ℱ1r0,Z0=1−μ−μ2r0Z0μ21+μ22,ℱ2r0,Z0=r0221+μ22+Z02μ4+1.The systemℱ1r0,Z0=ℱ2r0,Z0=0 has the zero r0∗,Z0∗=21+μ2,0 such that(97)det∂ℱ1,ℱ2∂r0,Z0|r0,Z0=r0∗,Z0∗=2μ2+μ−1μ21+μ22≠0.The proof of Corollary3 follows directly by applying Theorem 3 and (22) is obtained by substituting r0∗,Z0∗ in (20).


## 3.4. Proofs in Case4: λ≠0 and μ=δ≠0

### 3.4.1. Proof of Theorem4

Ifλ≠0 and μ=δ≠0, then equation (3) can be written as(98)x˙=y,y˙=z,z˙=u,u˙=v,v˙=λμ2x−μ2+2λμy+λ+2μ+λμ2z−μ2+2λμ+1u+λ+2μv+εFx,y,z,u,v.System (98) with ε=0 has a unique singular point at the origin and the linear part of this system has the eigenvalues ±i, λ, and μ. Using the change of variables(99)XYZUV=−λμ2μ2+2λμ−λ−2μ100λμ2−μ2−2λμλ+2μ−1μ2−2μμ2+1−2μ1−λ1−λ10λμ−λ−μ1+λμ−λ−μ1xyzuv,we transform the system (98) into the following system:(100)X˙=−Y,Y˙=X−εF˜X,Y,Z,U,V,Z˙=λZ+εF˜X,Y,Z,U,V,U˙=μU+V,V˙=μV+εF˜X,Y,Z,U,V,where(101)F˜=F˜X,Y,Z,U,V=Fd1,d2,d3,d4,d5,d1=−λμ2+2μ+λXλ2+1μ2+12+μ2+2λμ−1Yλ2+1μ2+12+Zλ2+1λ−μ2−Uμ2+1λ−μ−3μ2−2λμ+1Vμ2+12λ−μ2,d2=μ2+2λμ−1Xλ2+1μ2+12−−λμ2+2μ+λYλ2+1μ2+12+λZλ2+1λ−μ2−μUμ2+1λ−μ−2μ3−λμ2+λVμ2+12λ−μ2,d3=−−λμ2+2μ+λXλ2+1μ2+12−μ2+2λμ−1Yλ2+1μ2+12+λ2Zλ2+1λ−μ2−μ2Uμ2+1λ−μ−μμ3−μ+2λVμ2+12λ−μ2,d4=−μ2+2λμ−1Xλ2+1μ2+12+−λμ2+2μ+λYλ2+1μ2+12+λ3Zλ2+1λ−μ2−μ3Uμ2+1λ−μ−μ2λμ2−2μ+3λVμ2+12λ−μ2,d5=−λμ2+2μ+λXλ2+1μ2+12+μ2+2λμ−1Yλ2+1μ2+12+λ4Zλ2+1λ−μ2−μ4Uμ2+1λ−μ−μ3−μ3+2λμ2−3μ+4λVμ2+12λ−μ2.Note that the linear part of system (100) is in the real normal Jordan form of the linear part of system (98). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(102)r˙=−εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=λZ+εGr,θ,Z,U,V,U˙=μU+V,V˙=μV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(103)drdθ=−εsinθGr,θ,Z,U,V+oε2,dZdθ=λZ+ε1+λZrcosθGr,θ,Z,U,V+oε2,dUdθ=μU+V+εμU+VrcosθGr,θ,Z,U,V+oε2,dVdθ=μV+ε1+μVrcosθGr,θ,Z,U,V+oε2.System (103) is now of the same form as system (33) with(104)x=rZUV,t=θ,F0x,θ=0λZμU+VμV,F1x,θ=−sinθGr,θ,Z,U,V1+λZrcosθGr,θ,Z,U,VμU+VrcosθGr,θ,Z,U,V1+μVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (103). System (103) with ε=0 has the 2π−periodic solutions(105)rθZθUθVθ=r0000,∀r0>0.By the notations of Theorem6, we have that k=1 and n=4. Let r1>0 and r2>0; we take V=r1,r2⊂ℝ, α=r0∈r1,r2, and(106)β0:r1,r2⟶ℝ3,:r0↦β0r0=0,0,0.We also take(107)Z=zα=r0,0,0,0,r0∈r1,r2.The fundamental matrixMzαθ of the linear system (103) with ε=0 with respect to the periodic solution zα=r0,0,0,0 satisfying that Mzα0 is the identity matrix is(108)Mzαθ=10000eλθ0000eμθθeμθ000eμθ.We have(109)Mzα−10−Mzα−12π=000001−e−2πλ00001−e−2πμ2πe−2πμ0001−e−2πμ,which satisfy the assumption (ii) of Theorem 6. Taking(110)ξ:ℝ×ℝ3⟶ℝ,:r,Z,U,V↦ξr,Z,U,V=r,we must compute the function ℱα given by (34), and we obtain(111)ℱα=12π02π−sinθGr0,θ,0,0,0dθ,ℱr0=12π02π−sinθFD1,D2,D3,D4,D5dθ,and D1,D2,D3,D4, and D5 are given by (24). Then, by Theorem 6, for every simple zero r0∗ of the function ℱr0, there exists a limit cycle r,Z,U,Vθ,ε of system (103) such that(112)r,Z,U,V0,ε⟶r0∗,0,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (102) such that(113)r,θ,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system such that(114)X,Y,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (25) of equation (26) when ε⟶0.Theorem4 is proved.


### 3.4.2. Proof of Corollary4

IfFx,x˙,x¨,x⃛,x⃜=x3−x−1, then we have(115)ℱr0=r0−1+2λμ+μ241+λ21+μ22−3r0281+λ221+μ24,which have the real positive simple zero r0∗=23/31+λ21+μ2 with(116)dfdr0r0∗=1−2λμ−μ21+λ21+μ22≠0.The proof of Corollary4 follows directly by applying Theorem 4 and (27) is obtained by substituting r0∗ in (25).


## 3.4.1. Proof of Theorem4

Ifλ≠0 and μ=δ≠0, then equation (3) can be written as(98)x˙=y,y˙=z,z˙=u,u˙=v,v˙=λμ2x−μ2+2λμy+λ+2μ+λμ2z−μ2+2λμ+1u+λ+2μv+εFx,y,z,u,v.System (98) with ε=0 has a unique singular point at the origin and the linear part of this system has the eigenvalues ±i, λ, and μ. Using the change of variables(99)XYZUV=−λμ2μ2+2λμ−λ−2μ100λμ2−μ2−2λμλ+2μ−1μ2−2μμ2+1−2μ1−λ1−λ10λμ−λ−μ1+λμ−λ−μ1xyzuv,we transform the system (98) into the following system:(100)X˙=−Y,Y˙=X−εF˜X,Y,Z,U,V,Z˙=λZ+εF˜X,Y,Z,U,V,U˙=μU+V,V˙=μV+εF˜X,Y,Z,U,V,where(101)F˜=F˜X,Y,Z,U,V=Fd1,d2,d3,d4,d5,d1=−λμ2+2μ+λXλ2+1μ2+12+μ2+2λμ−1Yλ2+1μ2+12+Zλ2+1λ−μ2−Uμ2+1λ−μ−3μ2−2λμ+1Vμ2+12λ−μ2,d2=μ2+2λμ−1Xλ2+1μ2+12−−λμ2+2μ+λYλ2+1μ2+12+λZλ2+1λ−μ2−μUμ2+1λ−μ−2μ3−λμ2+λVμ2+12λ−μ2,d3=−−λμ2+2μ+λXλ2+1μ2+12−μ2+2λμ−1Yλ2+1μ2+12+λ2Zλ2+1λ−μ2−μ2Uμ2+1λ−μ−μμ3−μ+2λVμ2+12λ−μ2,d4=−μ2+2λμ−1Xλ2+1μ2+12+−λμ2+2μ+λYλ2+1μ2+12+λ3Zλ2+1λ−μ2−μ3Uμ2+1λ−μ−μ2λμ2−2μ+3λVμ2+12λ−μ2,d5=−λμ2+2μ+λXλ2+1μ2+12+μ2+2λμ−1Yλ2+1μ2+12+λ4Zλ2+1λ−μ2−μ4Uμ2+1λ−μ−μ3−μ3+2λμ2−3μ+4λVμ2+12λ−μ2.Note that the linear part of system (100) is in the real normal Jordan form of the linear part of system (98). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(102)r˙=−εsinθGr,θ,Z,U,V,θ˙=1+εrcosθGr,θ,Z,U,V,Z˙=λZ+εGr,θ,Z,U,V,U˙=μU+V,V˙=μV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(103)drdθ=−εsinθGr,θ,Z,U,V+oε2,dZdθ=λZ+ε1+λZrcosθGr,θ,Z,U,V+oε2,dUdθ=μU+V+εμU+VrcosθGr,θ,Z,U,V+oε2,dVdθ=μV+ε1+μVrcosθGr,θ,Z,U,V+oε2.System (103) is now of the same form as system (33) with(104)x=rZUV,t=θ,F0x,θ=0λZμU+VμV,F1x,θ=−sinθGr,θ,Z,U,V1+λZrcosθGr,θ,Z,U,VμU+VrcosθGr,θ,Z,U,V1+μVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (103). System (103) with ε=0 has the 2π−periodic solutions(105)rθZθUθVθ=r0000,∀r0>0.By the notations of Theorem6, we have that k=1 and n=4. Let r1>0 and r2>0; we take V=r1,r2⊂ℝ, α=r0∈r1,r2, and(106)β0:r1,r2⟶ℝ3,:r0↦β0r0=0,0,0.We also take(107)Z=zα=r0,0,0,0,r0∈r1,r2.The fundamental matrixMzαθ of the linear system (103) with ε=0 with respect to the periodic solution zα=r0,0,0,0 satisfying that Mzα0 is the identity matrix is(108)Mzαθ=10000eλθ0000eμθθeμθ000eμθ.We have(109)Mzα−10−Mzα−12π=000001−e−2πλ00001−e−2πμ2πe−2πμ0001−e−2πμ,which satisfy the assumption (ii) of Theorem 6. Taking(110)ξ:ℝ×ℝ3⟶ℝ,:r,Z,U,V↦ξr,Z,U,V=r,we must compute the function ℱα given by (34), and we obtain(111)ℱα=12π02π−sinθGr0,θ,0,0,0dθ,ℱr0=12π02π−sinθFD1,D2,D3,D4,D5dθ,and D1,D2,D3,D4, and D5 are given by (24). Then, by Theorem 6, for every simple zero r0∗ of the function ℱr0, there exists a limit cycle r,Z,U,Vθ,ε of system (103) such that(112)r,Z,U,V0,ε⟶r0∗,0,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (102) such that(113)r,θ,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system such that(114)X,Y,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (25) of equation (26) when ε⟶0.Theorem4 is proved.


## 3.4.2. Proof of Corollary4

IfFx,x˙,x¨,x⃛,x⃜=x3−x−1, then we have(115)ℱr0=r0−1+2λμ+μ241+λ21+μ22−3r0281+λ221+μ24,which have the real positive simple zero r0∗=23/31+λ21+μ2 with(116)dfdr0r0∗=1−2λμ−μ21+λ21+μ22≠0.The proof of Corollary4 follows directly by applying Theorem 4 and (27) is obtained by substituting r0∗ in (25).


## 3.5. Proofs in Case5: λ=μ=δ≠0

### 3.5.1. Proof of Theorem5

Ifλ=μ=δ≠0, then equation (3) can be written as(117)x˙=y,y˙=z,z˙=u,u˙=v,v˙=λ3x−3λ2y+3λ+λ3z−3λ2+1u+3λv+εFx,y,z,u,v.System (117) with ε=0 has a unique singular point at the origin and the linear part of this system has the eigenvalues ±i and λ. Using the change of variables(118)XYZUV=λ3−3λ23λ−100−λ33λ2−3λ110100−λ1−λ10λ2−2λλ2+1−2λ1xyzuv,we transform system (117) into the following system:(119)X˙=−Y,Y˙=X+εF˜X,Y,Z,U,V,Z˙=λZ+U,U˙=λU+V,V˙=λV+εF˜X,Y,Z,U,V,where(120)F˜=F˜X,Y,Z,U,V=Fe1,e2,e3,e4,e5,e1=λλ2−3X+1−3λ2Y+λ2+12Z−2λλ2+1U+3λ2−1Vλ2+13,e2=1−3λ2X−λλ2−3Y+λλ2+12Z+1−λ4U+λλ2−3Vλ2+13,e3=−λλ2−3X−1−3λ2Y+λ2λ2+12Z+2λλ2+1U+1−3λ2Vλ2+13,e4=−1−3λ2X+λλ2−3Y+λ3λ2+12Z+λ2λ2+3λ2+1U−λλ2−3Vλ2+13,e5=λλ2−3X+1−3λ2Y+λ4λ2+12Z+2λ3λ2+2λ2+1U+λ23λ2+λ4+6Vλ2+13.Note that the linear part of system (119) is in the real normal Jordan form of the linear part of system (51). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(121)r˙=εsinθGr,θ,Z,U,V,θ.=1+εrcosθGr,θ,Z,U,V,Z˙=λZ+U,U˙=λU+V,V˙=λV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(122)drdθ=εsinθGr,θ,Z,U,V+oε2,dZdθ=λZ+U−ελZ+UrcosθGr,θ,Z,U,V+oε2,dUdθ=λU+V−ελU+VrcosθGr,θ,Z,U,V+oε2,dVdθ=λV+ε1−λVrcosθGr,θ,Z,U,V+oε2.System (122) is now of the same form as system (33) with(123)x=rZUV,t=θ,F0x,θ=0λZ+UλU+VλV,F1x,θ=−sinθGr,θ,Z,U,V−λZ+UrcosθGr,θ,Z,U,V−λU+VrcosθGr,θ,Z,U,V1−λVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (122). System (122) with ε=0 has the 2π−periodic solutions(124)rθZθUθVθ=r0000,∀r0>0.By the notations of Theorem6, we have that k=1 and n=4. Let r1>0 and r2>0; we take V=r1,r2⊂ℝ, α=r0∈r1,r2, and(125)β0:r1,r2⟶ℝ3,:r0↦β0r0=0,0,0.We also take(126)Z=zα=r0,0,0,0,r0∈r1,r2.The fundamental matrixMzαθ of the linear system (122) with ε=0 with respect to the periodic solution zα=r0,0,0,0 satisfying that Mzα0 is the identity matrix is(127)Mzαθ=10000eλθθeλθθ22eλθ00eλθθeλθ000eλθ.We have(128)Mzα−10−Mzα−12π=000001−e−2πλ2πe−2πλ−2π2e−2πλ001−e−2πλ2πe−2πλ0001−e−2πλ,which satisfy the assumption (ii) of Theorem 6. Taking(129)ξ:ℝ×ℝ3⟶ℝ,:r,Z,U,V↦ξr,Z,U,V=r,we must compute the function ℱα given by (36), and we obtain(130)ℱr0=12π02πsinθGr0,θ,0,0,0dθ,ℱα=12π02πsinθFE1,E2,E3,E4,E5dθ,and E1,E2,E3,E4, and E5 are given by (29). Then, by Theorem 6, for every simple zero r0∗ of the function ℱr0, there exists a limit cycle r,Z,U,Vθ,ε of system (122) such that(131)r,Z,U,V0,ε⟶r0∗,0,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (121) such that(132)r,θ,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (119) such that(133)X,Y,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (30) of equation (31) when ε⟶0.Theorem5 is proved.


### 3.5.2. Proof of Corollary5

IfFx,x˙,x¨,x⃛,x⃜=−x¨2x⃜2+x3+x2−x˙x¨+x⃛−1, then we have(134)ℱr0=r03r023λ2−1−4λλ2−31+λ2381+λ26,which have the real positive simple zero r0∗=23/3λ3λ2−1λ2−31+λ21+λ2/3λ2−1 with(135)dfdr0r0∗=λλ2−31+λ23≠0.The proof of Corollary5 follows directly by applying Theorem 5 and (32) is obtained by substituting r0∗ in (30).


## 3.5.1. Proof of Theorem5

Ifλ=μ=δ≠0, then equation (3) can be written as(117)x˙=y,y˙=z,z˙=u,u˙=v,v˙=λ3x−3λ2y+3λ+λ3z−3λ2+1u+3λv+εFx,y,z,u,v.System (117) with ε=0 has a unique singular point at the origin and the linear part of this system has the eigenvalues ±i and λ. Using the change of variables(118)XYZUV=λ3−3λ23λ−100−λ33λ2−3λ110100−λ1−λ10λ2−2λλ2+1−2λ1xyzuv,we transform system (117) into the following system:(119)X˙=−Y,Y˙=X+εF˜X,Y,Z,U,V,Z˙=λZ+U,U˙=λU+V,V˙=λV+εF˜X,Y,Z,U,V,where(120)F˜=F˜X,Y,Z,U,V=Fe1,e2,e3,e4,e5,e1=λλ2−3X+1−3λ2Y+λ2+12Z−2λλ2+1U+3λ2−1Vλ2+13,e2=1−3λ2X−λλ2−3Y+λλ2+12Z+1−λ4U+λλ2−3Vλ2+13,e3=−λλ2−3X−1−3λ2Y+λ2λ2+12Z+2λλ2+1U+1−3λ2Vλ2+13,e4=−1−3λ2X+λλ2−3Y+λ3λ2+12Z+λ2λ2+3λ2+1U−λλ2−3Vλ2+13,e5=λλ2−3X+1−3λ2Y+λ4λ2+12Z+2λ3λ2+2λ2+1U+λ23λ2+λ4+6Vλ2+13.Note that the linear part of system (119) is in the real normal Jordan form of the linear part of system (51). We pass now from the Cartesian coordinates X,Y,Z,U,V to the cylindrical ones r,θ,Z,U,V with X=rcosθ, Y=rsinθ, and we obtain(121)r˙=εsinθGr,θ,Z,U,V,θ.=1+εrcosθGr,θ,Z,U,V,Z˙=λZ+U,U˙=λU+V,V˙=λV+εGr,θ,Z,U,V,where Gr,θ,Z,U,V=F˜rcosθ,rsinθ,Z,U,V.After dividing byθ. and simplifying, we find(122)drdθ=εsinθGr,θ,Z,U,V+oε2,dZdθ=λZ+U−ελZ+UrcosθGr,θ,Z,U,V+oε2,dUdθ=λU+V−ελU+VrcosθGr,θ,Z,U,V+oε2,dVdθ=λV+ε1−λVrcosθGr,θ,Z,U,V+oε2.System (122) is now of the same form as system (33) with(123)x=rZUV,t=θ,F0x,θ=0λZ+UλU+VλV,F1x,θ=−sinθGr,θ,Z,U,V−λZ+UrcosθGr,θ,Z,U,V−λU+VrcosθGr,θ,Z,U,V1−λVrcosθGr,θ,Z,U,V.We shall apply Theorem6 to system (122). System (122) with ε=0 has the 2π−periodic solutions(124)rθZθUθVθ=r0000,∀r0>0.By the notations of Theorem6, we have that k=1 and n=4. Let r1>0 and r2>0; we take V=r1,r2⊂ℝ, α=r0∈r1,r2, and(125)β0:r1,r2⟶ℝ3,:r0↦β0r0=0,0,0.We also take(126)Z=zα=r0,0,0,0,r0∈r1,r2.The fundamental matrixMzαθ of the linear system (122) with ε=0 with respect to the periodic solution zα=r0,0,0,0 satisfying that Mzα0 is the identity matrix is(127)Mzαθ=10000eλθθeλθθ22eλθ00eλθθeλθ000eλθ.We have(128)Mzα−10−Mzα−12π=000001−e−2πλ2πe−2πλ−2π2e−2πλ001−e−2πλ2πe−2πλ0001−e−2πλ,which satisfy the assumption (ii) of Theorem 6. Taking(129)ξ:ℝ×ℝ3⟶ℝ,:r,Z,U,V↦ξr,Z,U,V=r,we must compute the function ℱα given by (36), and we obtain(130)ℱr0=12π02πsinθGr0,θ,0,0,0dθ,ℱα=12π02πsinθFE1,E2,E3,E4,E5dθ,and E1,E2,E3,E4, and E5 are given by (29). Then, by Theorem 6, for every simple zero r0∗ of the function ℱr0, there exists a limit cycle r,Z,U,Vθ,ε of system (122) such that(131)r,Z,U,V0,ε⟶r0∗,0,0,0,whenε⟶0.Going back through the change of coordinates, we obtain a limit cycler,θ,Z,U,Vt,ε of system (121) such that(132)r,θ,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.We have a limit cycleX,Y,Z,U,Vt,ε of system (119) such that(133)X,Y,Z,U,V0,ε⟶r0∗,0,0,0,0,whenε⟶0.Finally, we obtain a limit cyclext,ε of equation (3) tending to the periodic solution (30) of equation (31) when ε⟶0.Theorem5 is proved.


## 3.5.2. Proof of Corollary5

IfFx,x˙,x¨,x⃛,x⃜=−x¨2x⃜2+x3+x2−x˙x¨+x⃛−1, then we have(134)ℱr0=r03r023λ2−1−4λλ2−31+λ2381+λ26,which have the real positive simple zero r0∗=23/3λ3λ2−1λ2−31+λ21+λ2/3λ2−1 with(135)dfdr0r0∗=λλ2−31+λ23≠0.The proof of Corollary5 follows directly by applying Theorem 5 and (32) is obtained by substituting r0∗ in (30).


## 4. Conclusion

There are several theories and methods for the study of the existence, uniqueness, or number and stability of limit cycles of differential equations which have been developed in trying to answer Hilbert’s sixteenth problem posed in 1900 (see reference [1]) about the maximum number of limit cycles that a planar polynomial differential system can obtain. In this work, we study the limit cycles of the fifth-order differential equation by using the averaging theory of first order [6, 7], and we provide sufficient conditions for the existence of limit cycles of equation (12); in the next work, we will try to apply the same method on higher order differential equations.



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