Periodic solutions for a class of fifth-order differential equations

Purpose – This study aims to provide sufficient conditions for the existence of periodic solutions of the fifth- order differential equation. Design/methodology/approach – Theauthorsshallusetheaveragingtheory,morepreciselyTheorem$6$. Findings – The main results on the periodic solutions of the fifth-order differential equation (equation (1)) are given in the statement of Theorem 1 and 2. Originality/value – In this article, the authors provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.


Introduction and statement of the main results
One of the main problems in the theory of differential equations is the study of their periodic orbits, their existence, their number and their stability. The goal of this paper is to study the periodic solutions of the fifth-order non-autonomous differential equation: where λ and ε are real parameters; p is a rational number different from −1, 0, 1, ε is sufficiently small; and F is a nonlinear non-autonomous periodic function. There are many papers studying the periodic orbits of fifth-order differential equations, see for instance in Refs. [1][2][3][4][5][6]. But, our main tool for studying the periodic orbits of equation (1) is completely different from the tools mentioned papers, and consequently, the results obtained seem distinct and new. We shall use the averaging theory, more precisely Theorem 5. Many of the quoted papers dealing with the periodic orbits of fifth-order differential equations use Schauder's or Leray-Schauder's fixed point theorem, the non-local reduction method or variational methods. In Refs. [7][8][9], the authors studied the limit cycles of the fourth-, sixth-and eighth-order non-autonomous differential equations.
In general, to obtain analytically periodic solutions of a differential system is a very difficult task, usually impossible. Here, with the averaging theory, this difficult problem for the differential equation (1) is reduced to find the zeros of a nonlinear function. We must say that the averaging theory for finding periodic solutions in general does not provide all the periodic solutions of the system. For more information about the averaging theory, see Section 2 and the references quoted there.
Our main results on the periodic solutions of the fifth-order differential equation ð1Þ are the following.
Theorem 1. Assume that p ¼ m=n is a rational different from −1, 0; 1, λ ≠ 0 in differential equation ð1Þ. Let (2) where m; n are positive integers, and If the function F is 2πn − periodic with respect to the variable t, then for every X * 0 ; Y * 0 ; Z * 0 ; U * 0 solution of the system: the differential equation ð1Þ has a periodic solution xðt; εÞ tending to the solution x 0 ðtÞ given by: Note that this solution is periodic of period 2πn. Theorem 1 is proved in Section 3. Its proof is based on the averaging theory for computing periodic orbits, see Section 2.
Theorem 3. Assume that p ¼ m=n is a rational different from −1, 0; 1, λ ¼ 0 in differential equation (1). Let Fðt; A; B; C; D; J Þdt; If the function F is 2πn − periodic with respect to the variable t, then for every the differential equation ð1Þ has a periodic solution xðt; εÞ tending to the solution x 0 ðtÞ given by: Note that this solution is periodic of period 2πn. Theorem 5 is proved in Section 4. Its proof is based on the averaging theory for computing periodic orbits, see Section 2. An application of Theorem 3 is given in the following corollary: : ; x :: ; x ::: ; x :::: (1) with p ¼ 1 2 , λ ¼ 0 has six periodic solutions x k ðt; εÞ for k ¼ 1; :::; 6 tending to the periodic solutions: Corollary 4 is proved in Section 5.

Basic results on the averaging theory
In this section, we present the basic results from the averaging theory that we shall need for proving the main results of this paper. We consider the problem of the bifurcation of T-periodic solutions from differential systems of the form: x $ ¼ F 0 ðt; xÞ þ εF 1 ðt; xÞ þ ε 2 F 2 ðt; x; εÞ; (11) with ε > 0 sufficiently small. Here the functions F 0 ; and Ω is an open subset of R n . The main assumption is that the unperturbed system: has a submanifold of periodic solutions. A solution of this problem is given using the averaging theory.
Let xðt; z; εÞ be the solution of the system ð12Þ such that xð0; z; εÞ ¼ z. We write the linearization of the unperturbed system along a periodic solution xðt; z; 0Þ as: In what follows, we denote by M z ðtÞ a fundamental matrix of the linear differential system (13), and by ξ : R k 3 R n−k → R k the projection of R n onto its first k coordinates, i.e. ξðx 1 ; :::; x n Þ ¼ ðx 1 ; :::; x k Þ. We assume that there exists a k-dimensional submanifold Z of Ω filled with T-periodic solutions of (12). Then, an answer to the problem of bifurcation of T-periodic solutions from the periodic solutions contained in Z for system (11) is given in the following result.
Theorem 5. Let W be an open and bounded subset of R k , and let β : CLðW Þ → R n−k be a C 2 function. We assume that: and that for each z α ∈ Z, the solution xðt; z α Þ of (12) is T-periodic; (2) For each z α ∈ Z, there is a fundamental matrix M zα ðtÞ of (13) such that the matrix M −1 zα ð0Þ − M −1 zα ðTÞ has in the upper-right corner the k 3 ðn − kÞ zero matrix, and in the lower-right corner a ðn − kÞ 3 ðn − kÞ matrix Δ α with detðΔ α Þ ≠ 0.
Theorem 5 goes back to Malkin [10] and Roseau [11]; for a shorter proof, see Ref. [12]. We assume that there exists an open set V with CLðV Þ ⊂ Ω such that for each z ∈ CLðV Þ, xðt; z; 0Þ is T-periodic, where xðt; z; 0Þ denotes the solution of the unperturbed system (12) with xð0; z; 0Þ ¼ z. The set CLðV Þ is isochronous for the system (11), i.e. it is a set formed only by periodic orbits, all of them having the same period. Then, an answer to the problem of the bifurcation of T-periodic solutions from the periodic solutions xðt; z; 0Þ contained in CLðV Þ is given in the following result.
For a shorter proof of Theorem 6, see Corollary 1 of [12]. In fact, this result goes back to Malkin [10] and Roseau [11].

Proof of Theorem 1
, u ¼ x ::: , v ¼ x :::: , then system (1) can be written as: Fifth-order differential equations The unperturbed system has a unique singular point, the origin. The eigenvalues of the linearized system at this singular point are ±i, ±pi and λ. By the linear invertible transformation: ðX ; Y ; Z ; U ; V Þ T ¼ Bðx; y; z; u; vÞ T ; where we transform the system (16) such that its linear part is real Jordan normal form of the linear part of system (16) with ε ¼ 0, i.e.: Note that the linear part of the differential system (18) at the origin is in its real Jordan normal form, and that the change of variables (17) is defined when p is a rational different from −1, 0; 1, because the determinant of the matrix of the change is pðp 2 − 1Þ 2 ðλ 2 þ 1Þðλ 2 þ p 2 Þ.
We shall apply Theorem 5 to the differential system (18). We note that system (18) can be written as system 11 ð Þ taking We shall study the periodic solutions of system (18) in our case, i.e. the periodic solutions of system (18) with ε ¼ 0. These periodic solutions are:

AJMS
This set of periodic orbits has dimension four, all having the same period 2πn, where n is defined in the statement of Theorem 1. To look for the periodic solutions of our equation (1) we must calculate the zeros z ¼ ðX 0 ; Y 0 ; Z 0 ; U 0 ; V 0 Þ of the system F ðzÞ ¼ 0, where F ðzÞ is given by (14). The fundamental matrix M ðtÞ of the differential system (18) with ε ¼ 0, along any periodic solution is: Moreover, an easy computation shows that: We obtain ð1 − expð−2πnλÞÞ ≠ 0, because λ ≠ 0. Consequently, all the assumptions of Theorem 5 are satisfied. Therefore, we must study the zeros in W of the system F ðzÞ ¼ 0 of four equations with four unknowns, where W and F are given in the statement of Theorem 5. More precisely, we have F ðzÞ ¼ ðF 1 ðzÞ; F 2 ðzÞ; F 3 ðzÞ; F 4 ðzÞÞ, such that z ¼ ðX 0 ; Y 0 ; Z 0 ; U 0 Þ, where the functions F 1 , F 2 ; F 3 and F 4 are the ones given in (2). The zeros X * 0 ; Y * 0 ; Z * 0 ; U * 0 of system 4 ð Þ with respect to the variables X 0 , Y 0 , Z 0 and U 0 provide periodic orbits of system 18 ð Þ with ε ≠ 0 sufficiently small if they are simple, i.e. if the condition 5 ð Þ is satisfied. Going back through the change of variables, for every simple zero Note that this solution is periodic of period 2πn.
This completes the proof of Theorem 1.

Proof of Theorem 3
We want to study the periodic orbits of the class of fifth-order differential equation: This is the case of equation (1) when λ ¼ 0, and p is a rational number different from −1, 0; 1.

Fifth-order differential equations
If y ¼ x : , z ¼ x :: , u ¼ x ::: , v ¼ x :::: , we write the fifth-order differential equation (19) as the following first-order differential system: The unperturbed system has a unique singular point, the origin. The eigenvalues of the linearized system at this singular point are ±i, ±pi and 0. By the linear invertible transformation: we transform the system 20 ð Þsuch that its linear part is real Jordan normal form of the linear part of system 20 ð Þ with ε ¼ 0, i.e.: with A; B; C; D,and J as in the statement of Theorem 3. Note that the linear part of the differential system (21) at the origin is in its real Jordan normal form. We shall apply Theorem 6 to the differential system (21). We note that system (21) can be written as system 11 ð Þ taking

AJMS
We shall study the periodic solutions of system 21 ð Þin our case, i.e. the periodic solutions of system (21) with ε ¼ 0. These periodic solutions are: This set of periodic orbits has dimension five, all having the same period 2πn, where n is defined in the statement of Theorem 3. To look for the periodic solutions of our equation (19), we must calculate the zeros Now computing the function F ðzÞ given in 15 ð Þ, we got that the system F ðzÞ ¼ 0, can be written as system 8 ð Þ with the function F k ðX 0 ; Y 0 ; Z 0 ; U 0 ; V 0 Þ given in 6 ð Þ. The zeros X * 0 ; Y * 0 ; Z * 0 ; U * 0 ; V * 0 of system 8 ð Þ with respect to the variables X 0 , Y 0 , Z 0 , U 0 , and V 0 , provide periodic orbits of system 21 ð Þ with ε ≠ 0 sufficiently small if they are simple, i.e. if 9 ð Þ holds. Going back through the change of variables, for every simple zero X * 0 ; Y * 0 ; Z * 0 ; U * 0 ; V * 0 of system 8 ð Þ, we obtain a 2πn periodic solution xðtÞ of the differential equation (1) for ε ≠ 0 sufficiently small such that xðtÞ tends to the periodic solution 10 ð Þof x ð5Þ þ ðp 2 þ 1Þx ... þp 2 x $ ¼ 0 when ε → 0. Note that this solution is periodic of period 2πn. This completes the proof of Theorem 3.

Proof of Corollaries 2 and 4
Proof of Corollary 2. Consider the function Respectively, we obtain using Theorem 3 the ten solutions given in statement of the corollary. ,