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

Department of Mathematics, University of Guelma, BO 401, Guelma 24000, Algeria Department of Mathematics, College of Sciences and Arts in Arrass, Qassim University, Buraydah, Saudi Arabia Preparatory Institute for Engineering Studies in Sfax, Sfax, Tunisia College of Industrial Engineering, King Khalid University, Abha, Saudi Arabia Department of Mathematics, College of Sciences, Juba University, Juba, Sudan


Introduction and Statement of the
Main Results e 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. ere 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.
e 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][3][4][5][6][7][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 with μ ≠ 0; ε is a small real parameter; F ∈ C 2 is 2π− periodic in t.
In [6], the authors studied the following fourth-order differential equation: where λ and μ are real, ε is a small real parameter, and F ∈ C 2 is 2π− periodic in t.
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: where a � λμδ, b � − (λμ + λδ + μδ), c � λ + μ + δ + λμδ, d � − (1 + λμ + λδ + μδ), e � λ + μ + δ, (4) where the dot means derivative with respect to an independent variable t, ε is a small enough parameter, and F ∈ C 2 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 x ⃜ , 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 parameters λ, μ, and δ, we distinguish the five following cases.  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.
Theorem 1. Assume that λμδ ≠ 0 and λ ≠ μ ≠ δ. For every positive simple zero r * 0 of the function F(r 0 ) given by (5) there is a limit cycle x(t, ε) of equation (3) tending to the periodic solution of when ε ⟶ 0. eorem 1 will be proved in Section 3.1.1. An application of eorem 1 is the following.
en, there is a limit cycle x 1 (t, ε) of equation (3) tending to the periodic solution of equation (8) when ε ⟶ 0. Corollary 1 will be proved in Section 3.1.2.
2 Discrete Dynamics in Nature and Society 1.2. Case 2: λ � 0, μ δ ≠ 0, and μ ≠ δ. We define the functions and Our main result for this case is the following theorem.
, then there is a limit cycle x 2 (t, ε) of equation (3) tending to the periodic solution of equation (15) when ε ⟶ 0. Corollary 2 will be proved in Section 3.
1.3. Case 3: λ � 0 and μ � δ ≠ 0. We define the functions where Our main result for this case is the following theorem.
x − 1, then there is a limit cycle x 3 (t, ε) of equation (3) tending to the periodic solution of equation (21) when ε ⟶ 0. Corollary 3 will be proved in Section 3.3.2.
Discrete Dynamics in Nature and Society 1.4. Case 4: λ ≠ 0 and μ � δ ≠ 0. We define the function where Our main result for this case is the following theorem.

Theorem 4.
Assume that λ ≠ 0 and δ � μ ≠ 0. For every positive simple zero r * 0 of the function F(r 0 ) given by (23), there is a limit cycle x(t, ε) of equation (3) tending to the periodic solution when ε ⟶ 0. eorem 4 will be proved in Section 3.4.1. An application of eorem 4 is the following.
x 4 (t, ε) of equation (3) tending to the periodic solution where Our main result for this case is the following theorem.
Theorem 5. Assume that μ � δ � λ ≠ 0. For every positive simple zero r * 0 of the function F(r 0 ) given by (28), there is a limit cycle x(t, ε) of equation (3) tending to the periodic solution of when ε ⟶ 0. eorem 5 will be proved in Section 3.5.1. An application of eorem 5 is the following.
, then there is a limit cycle x 5 (t, ε) of equation (3) tending to the periodic solution of equation (31) when ε ⟶ 0. Corollary 5 will be proved in Section 3.5.2.

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 of T− periodic solutions from the differential system and Ω is an open subset of R n . We suppose that the unperturbed system has a k-dimensional submanifold Z of periodic solutions. Let x(t, z) be the solution of the unperturbed system (34) such that x(0, z) � z. e linearisation of system (34) along the periodic solution x(t, z) is written as We denote by M z (t) some fundamental matrices of the linear differential system (35) and by ξ: R k × R n− k ⟶ R k the projection of R n onto its first k coordinates; i.e., ξ(x 1 , . . . , x n ) � (x 1 , . . . , x k ).

Theorem 6. Let V ⊂ R k be open and bounded and
has in the upper right corner the k × (n − k) zero matrix, and in the lower right corner a matrix eorem 1 goes back to [10] and [11]; for a shorter proof, see [12].
Note that the periodic orbits provided by eorem 6 are limit cycles.

Proofs of the Results in Case
and v � x ⃜ , then equation (3) can be written as 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 Discrete Dynamics in Nature and Society we transform system (37) into the following system: , , , 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 � r cos θ, Y � r sin θ, and we obtain After dividing by θ . and simplifying, we find System (42) is now of the same form as system (33) with 6 Discrete Dynamics in Nature and Society We shall apply eorem 6 to system (42). System (42) with ε � 0 has the 2π− periodic solutions By the notations of eorem 6, we have that k � 1 and n � 4. Let r 1 > 0 and r 2 > 0; we take V �]r 1 , r 2 [ ⊂ R, α � r 0 ∈ [r 1 , r 2 ], and : r 0 ↦β 0 r 0 � (0, 0, 0).
which have the real positive simple zero Discrete Dynamics in Nature and Society e proof of Corollary 1 follows directly by applying eorem 1 and (10) is obtained by substituting r * 0 in (7).  (3) can be written as y, z, u, v).

Proofs of the Results in
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 we transform system (56) into the following system: 8 Discrete Dynamics in Nature and Society 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 � r cos θ, Y � r sin θ, and we obtain _ r � ε sin θ G(r, θ, Z, U, V), where G(r, θ, Z, U, V) � F(r cos θ, r sin θ, Z, U, V).

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Discrete Dynamics in Nature and Society We also take e fundamental matrix M z α (θ) of the linear system (61) with ε � 0 with respect to the periodic solution z α � (r 0 , Z 0 , 0, 0) satisfying that M z α (0) is the identity matrix is We have which satisfy the assumption (ii) of eorem 6. Taking we must compute the function F(α) given by (36), and we obtain where and B 1 , B 2 , B 3 , B 4 , and B 5 are given by (12). en, by eorem 6, for every simple zero (r * 0 , Z * 0 ) of the function F(r 0 , Z 0 ) there exists a limit cycle (r, Z, U, V)(θ, ε) of system (61) such that Going back through the change of coordinates, we obtain a limit cycle (r, θ, Z, U, V)(t, ε) of system (60) such that We have a limit cycle (X, Y, Z, U, V)(t, ε) of system (58) such that Finally, we obtain a limit cycle x(t, ε) of equation (3) tending to the periodic solution (14) of equation (15) when ε ⟶ 0. eorem 2 is proved.

Proofs in
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 we transform system (77) into the following system: 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 � r cos θ, Y � r sin θ, and we obtain _ r � − ε sin θ G(r, θ, Z, U, V), where G(r, θ, Z, U, V) � F(r cos θ, r sin θ, Z, U, V). After dividing by θ . and simplifying, we find System (82) is now of the same form as system (33) with We shall apply eorem 6 to system (82). System (82) with ε � 0 has the 2π− periodic solutions Discrete Dynamics in Nature and Society By the notations of eorem 6, we have that k � 2 and n � 4. Let R > 0; we take : r 0 , Z 0 ↦β 0 r 0 , Z 0 � (0, 0).

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We also take e fundamental matrix M z α (θ) of the linear system (82) with ε � 0 with respect to the periodic solution z α � (r 0 , Z 0 , 0, 0) satisfying that M z α (0) is the identity matrix is We have which satisfy the assumption (ii) of eorem 6. Taking we must compute the function F(α) given by (36), and we obtain where and C 1 , C 2 , C 3 , C 4 , and C 5 are given by (18). en, by eorem 6, for every simple zero (r * 0 , Z * 0 ) of the function F(r 0 , Z 0 ), there exists a limit cycle (r, Z, U, V)(θ, ε) of system (82) such that Going back through the change of coordinates, we obtain a limit cycle (r, θ, Z, U, V)(t, ε) of system (81) such that We have a limit cycle (X, Y, Z, U, V)(t, ε) of system (79) such that Finally, we obtain a limit cycle x(t, ε) of equation (3) tending to the periodic solution (20) of equation (21) when ε ⟶ 0. eorem 3 is proved.

3.5.2.
Proof of which have the real positive simple zero e proof of Corollary 5 follows directly by applying eorem 5 and (32) is obtained by substituting r * 0 in (30).

Conclusion
ere 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.

Data Availability
No data were used to support the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.