Periodic solutions for a class of fifth-order differential equations

Abstract

Purpose

This study aims to provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.

Design/methodology/approach

The authors shall use the averaging theory, more precisely Theorem $6$.

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.

Citation

Berrehail, C.E. and Bouslah, Z. (2023), "Periodic solutions for a class of fifth-order differential equations", Arab Journal of Mathematical Sciences, Vol. 29 No. 1, pp. 2-13. https://doi.org/10.1108/AJMS-07-2020-0024

1. 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:

(1)x(5)−λx⃜+(p2+1)x…−λ(p2+1)x¨+p2x⋅−λp2x=εF(t,x,x⋅,x¨,x…,x.…),

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. [1–6]. 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. [7–9], 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 (1) are the following.

Theorem 1.

Assume that p=m/n is a rational different from −1, 0,1, λ≠0 in differential equation (1). Let

(2) F1(X0,Y0,Z0,U0)=12πn∫02πn⁡cos(t)F(t,A,B,C,D,J)dt,F2(X0,Y0,Z0,U0)=−12πn∫02πn⁡sin(t)F(t,A,B,C,D,J)dt,F3(X0,Y0,Z0,U0)=12πn∫02πn⁡cos(pt)F(t,A,B,C,D,J)dt,F4(X0,Y0,Z0,U0)=−12πn∫02πn⁡sin(pt)F(t,A,B,C,D,J)dt,

where m,n are positive integers, and

(3) A=−(X0+λY0)cos⁡t+(λX0−Y0)sin⁡t(p2−1)(λ2+1)+(pZ0+λU0)cos(pt)−(pU0−λZ0)sin(pt)p(p2−1)(λ2+p2)B=−(λX0−Y0)cos⁡t+(X0+λY0)sin⁡t(p2−1)(λ2+1)−(pU0−λZ0)cos(pt)+(λU0+pZ0)sin(pt)(p2−1)(λ2+p2)C=(X0+λY0)cos⁡t+(λX0−Y0)sin⁡t(p2−1)(λ2−1)+−(pZ0+λU0)p⁡cos(pt)+(pU0−λZ0)p⁡sin(pt)(p2−1)(λ2+p2)D=(λX0−Y0)cos⁡t−(X0+λY0)sin⁡t(p2−1)(λ2+1)+(pU0−λZ0)p2⁡cos(pt)+(λU0+pZ0)p2⁡sin(pt)(p2−1)(λ2+p2)J=−(X0+λY0)cos⁡t+(λX0−Y0)sin⁡t(p2−1)(λ2−1)+(pZ0+λU0)p3⁡cos(pt)−(pU0−λZ0)p3⁡sin(pt)(p2−1)(λ2+p2),

If the function F is 2πn−periodic with respect to the variable t, then for every (X0*,Y0*,Z0*,U0*) solution of the system:

(4)Fk(X0,Y0,Z0,U0)=0, k=1,...,4,

satisfying

(5)det∂(F2,F2,F3,F4)∂(X0,Y0,Z0,U0)|(X0,Y0,Z0,U0)=(X0*,Y0*,Z0*,U0*)≠0,

the differential equation (1) has a periodic solution x(t,ε) tending to the solution x0(t) given by:

−(X0*+λY0*)cos(t)+(λX0*−Y0*)sin(t)(p2−1)(λ2+1)+(pZ0*+λU0*)cos(pt)−(pU0*−λZ0*)sin(pt)p(p2−1)(λ2+p2)

of x(5)−λx….+(p2+1)x…−λ(p2+1)x¨+p2x⋅−λp2x=0 when ε→0. 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.

An application of Theorem 1 is the following.

Corollary 2.

If F(t,x,x.,x..,x...,x....)=(1+cos⁡t)(ax2+bx⋅) with a⋅b≠0, then the differential equation (1) with p=12, λ=2 has one periodic solution x2(t,ε) tending to the periodic solution x2(t) given by:

x2(t)=−2basin(t),

of x(5)−2x….+54x…−52x¨+14x⋅−12x=0 when ε→0.

Corollary 2 is proved in Section 5.

Theorem 3.

Assume that p=m/n is a rational different from −1, 0,1, λ=0 in differential equation (1). Let

(6) F1(X0,Y0,Z0,U0)=12πn∫02πn⁡cos(t)F(t,A,B,C,D,J)dt,F2(X0,Y0,Z0,U0)=−12πn∫02πn⁡sin(t)F(t,A,B,C,D,J)dt,F3(X0,Y0,Z0,U0)=12πn∫02πn⁡cos(pt)F(t,A,B,C,D,J)dt,F4(X0,Y0,Z0,U0)=−12πn∫02πn⁡sin(pt)F(t,A,B,C,D,J)dt,F5(X0,Y0,Z0,U0)=12πn∫02πnF(t,A,B,C,D,J)dt,

and

(7) A=−(X0⁡cos⁡t−Y0⁡sin⁡t)p2+Z0⁡cos(pt)−U0⁡sin(pt)+(p2−1)V0p2(p2−1),B=(Y0⁡cos⁡t+X0⁡sin⁡t)p−U0⁡cos(pt)−Z0⁡sin(pt)p(p2−1),C=X0⁡cos⁡t−Y0⁡sin⁡t−Z0⁡cos(pt)+U0⁡sin(pt)p2−1,D=−Y0⁡cos⁡t−X0⁡sin⁡t+p(U0⁡cos(pt)+Z0⁡sin(pt))p2−1,J=−X0⁡cos⁡t+Y0⁡sin⁡t+p2(Z0⁡cos(pt)−U0⁡sin(pt))(p2−1),

If the function F is 2πn−periodic with respect to the variable t, then for every (X0*,Y0*,Z0*,U0*,V0*) solution of the system:

(8)Fk(X0,Y0,Z0,U0,V0)=0, k=1,...,5,

satisfying

(9)det∂(F2,F2,F3,F4)∂(X0,Y0,Z0,U0,V0)|(X0,Y0,Z0,U0,V0)=(X0*,Y0*,Z0*,U0*,V0*)≠0,

the differential equation (1) has a periodic solution x(t,ε) tending to the solution x0(t) given by:

(10)−(X0*⁡cos⁡t−Y0*⁡sin⁡t)p2+Z0*⁡cos(pt)−U0*⁡sin(pt)+(p2−1)V0*p2(p2−1)

of x(5)−λx….+(p2+1)x…−λ(p2+1)x¨+p2x⋅−λp2x=0 when ε→0. 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:

Corollary 4.

If F(t,x,x.,x..,x...,x....)=(2x2−x⋅2+x−2x⋅)sin⁡t then the differential Eqn (1) with p=12, λ=0 has six periodic solutions xk(t,ε) for k=1,...,6 tending to the periodic solutions:

x1(t)=−14sin⁡t−12842cos(12t)−14, x2(t)=−14sin⁡t+12842cos(12t)−14,x3(t)=14sin⁡t+12842sin(12t)−14, x4(t)=14sin⁡t−12842sin(12t)−14,x5(t)=−11010sin⁡t−14, x6(t)=11010sin⁡t−14,

of x(5)+54x…+14x⋅=0 when ε→0.

Corollary 4 is proved in Section 5.

2. 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:

(11)x⋅=F0(t,x)+εF1(t,x)+ε2F2(t,x,ε),

with ε>0 sufficiently small. Here the functions F0,F1:R×Ω→Rn and F2:R×Ω×(−ε0,ε0)→Rn are C2 functions, T-periodic in the variable t, and Ω is an open subset of Rn. The main assumption is that the unperturbed system:

(12)x⋅=F0(t,x),

has a submanifold of periodic solutions. A solution of this problem is given using the averaging theory.

Let x(t,z,ε) be the solution of the system (12) such that x(0,z,ε)=z. We write the linearization of the unperturbed system along a periodic solution x(t,z,0) as:

(13)y⋅=DxF0(t,x(t,z,0))y.

In what follows, we denote by Mz(t) a fundamental matrix of the linear differential system (13), and by ξ:Rk×Rn−k→Rk the projection of Rn onto its first k coordinates, i.e. ξ(x1,...,xn)=(x1,...,xk).

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 Rk, and let β:CL(W)→Rn−k be a C2 function. We assume that:

1. Z=zα=α,β(α),α∈CL(W)⊂Ω, and that for each zα∈Z, the solution x(t,zα) of (12) is T-periodic;

2. For each zα∈Z, there is a fundamental matrix Mzα(t) of (13) such that the matrix Mzα−1(0)−Mzα−1(T) has in the upper-right corner the k×(n−k) zero matrix, and in the lower-right corner a (n−k)×(n−k) matrix Δα with det(Δα)≠0.

We consider the function F:CL(W)→Rk

(14)F(α)=ξ1T∫0TMzα−1(t)F1(t,x(t,zα))dt.

If there exists a∈W with F(a)=0 and detdF/dα(a)≠0, then there is a T-periodic solution φ(t,ε) of system (11) such that φ(0,ε)→za as ε→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 CL(V)⊂Ω such that for each z∈CL(V), x(t,z,0) is T-periodic, where x(t,z,0) denotes the solution of the unperturbed system (12) with x(0,z,0)=z. The set CL(V) 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 x(t,z,0) contained in CL(V) is given in the following result.

Theorem 6.

[Perturbations of an isochronous set] We assume that there exists an open and bounded set V with CL(V)⊂Ω such that for each z∈CL(V), the solution x(t,z) is T-periodic, then we consider the function F:CL(V)→Rn

(15) F(z)=∫0TMz−1(t,z)F1(t,x(t,z))dt.

If there exists a∈V with detdF/dα(a)≠0, then there exists a T-periodic solution φ(t,ε) of system (11) such that φ(0,ε)→a as ε→0.

For a shorter proof of Theorem 6, see Corollary 1 of [12]. In fact, this result goes back to Malkin [10] and Roseau [11].

3. Proof of Theorem 1

If y=x., z=x.., u=x..., v=x...., then system (1) can be written as:

x.=y,

y.=z,

z.=u,

u.=v,

(16)v.=λp2x−p2y+λ(p2+1)z−(p2+1)u+λv+εF(t,x,x⋅,x¨,x…,x….),

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:

(17)(X,Y,Z,U,V)T=B(x,y,z,u,v)T,

where

B=0−λp2p2−λ1−λp2p2−λ100−λ1−λ1−λpp−λpp0p20p2+101,

we transform the system (16) such that its linear part is real Jordan normal form of the linear part of system (16) with ε=0, i.e.:

(18){X.=−Y+εG(t,X,Y,Z,U,V),Y.=X,Z.=−pU+εGt,X,Y,Z,U,V,U.=pZ,V.=λV+εGt,X,Y,Z,U,V,

where

G=F(t,A,B,C,D,J)=Gt,X,Y,Z,U,V,

with A,B,C,D,and J as in the statement of Theorem 1.

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 p(p2−1)2(λ2+1)(λ2+p2).

We shall apply Theorem 5 to the differential system (18). We note that system (18) can be written as system 11 taking

x=XYZUV, F0x,t=−YX−pUpZλV, F1x,t=G0G0G.

We shall study the periodic solutions of system (18) in our case, i.e. the periodic solutions of system (18) with ε=0. These periodic solutions are:

XtYtZtUtV(t)=X0⁡cos(t)−Y0⁡sin(t)Y0⁡cos(t)+X0⁡sin(t)Z0⁡cos(pt)−U0⁡sin(pt)U0⁡cos(pt)+Z0⁡sin(pt)0.

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=(X0,Y0,Z0,U0,V0) of the system F(z)=0, where F(z) is given by (14). The fundamental matrix M(t) of the differential system (18) with ε=0, along any periodic solution is:

M(t)=Mz(t)=cos(t)−sin(t)000sin(t)cos(t)00000cos(pt)−sin(pt)000sin(pt)cos(pt)00000eλt.

The inverse matrix of Mt is:

MIt=cos(t)sin(t)000−sin(t)cos(t)00000cos(pt)sin(pt)000−sin(pt)cos(pt)00000e−λt.

Moreover, an easy computation shows that:

MI(0)−MI(2πn)=0000000000000000000000001−e−2πnλ.

We obtain (1−exp(−2πnλ))≠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(z)=0 of four equations with four unknowns, where W and F are given in the statement of Theorem 5. More precisely, we have F(z)=(F1(z),F2(z),F3(z),F4(z)), such that z=(X0,Y0,Z0,U0), where the functions F1, F2,F3 and F4 are the ones given in (2). The zeros (X0*,Y0*,Z0*,U0*) of system 4 with respect to the variables X0, Y0, Z0 and U0 provide periodic orbits of system 18 with ε≠0 sufficiently small if they are simple, i.e. if the condition 5 is satisfied. Going back through the change of variables, for every simple zero (X0*,Y0*,Z0*,U0*)∈R4−{(0,0,0,0)} of system 4, we obtain a 2πn periodic solution x(t) of the differential equation 1 for ε≠0 sufficiently small such that x(t) tends to the periodic solution, where x(t) is defined in the statement of Theorem 1, of x(5)−λx….+(p2+1)x…−λ(p2+1)x¨+p2x⋅−λp2x=0 when ε→0. Note that this solution is periodic of period 2πn. This completes the proof of Theorem 1.

4. Proof of Theorem 3

We want to study the periodic orbits of the class of fifth-order differential equation:

(19)x(5)+(p2+1)x…+p2x⋅=εF(t,x,x⋅,x¨,x…,x….).

This is the case of equation (1) when λ=0, and p is a rational number different from −1, 0,1.

If y=x., z=x.., u=x..., v=x...., we write the fifth-order differential equation (19) as the following first-order differential system:

x.=y,

y.=z,

z.=u,

u.=v,

(20)v.=−p2y−(p2+1)u+εF(t,x,x⋅,x¨,x…,x….),

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:

(X,Y,Z,U,V)T=B(x,y,z,u,v)T,

where

B=00p2010p2010001010p0p0p20p2+101,

we transform the system 20 such that its linear part is real Jordan normal form of the linear part of system 20 with ε=0, i.e.:

(21){X.=−Y+εGt,X,Y,Z,U,V,Y.=X,Z.=−pU+εGt,X,Y,Z,U,V,U.=pZ,V.=εGt,X,Y,Z,U,V,

where

G=F(t,A,B,C,D,J)=Gt,X,Y,Z,U,V,

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 taking

x=XYZUV, F0x,t=−YX−pUpZ0, F1x,t=G0G0G.

We shall study the periodic solutions of system 21 in our case, i.e. the periodic solutions of system (21) with ε=0. These periodic solutions are:

XtYtZtUtV(t)=X0⁡cos⁡t−Y0⁡sin⁡tY0⁡cos⁡t+X0⁡sin⁡tZ0⁡cos(pt)−U0⁡sin(pt)U0⁡cos(pt)+Z0⁡sin(pt)V0.

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 z=(X0,Y0,Z0,U0,V0) of the system F(z)=0, where F(z) is given by 15. The fundamental matrix M(t) of the differential system (21) with ε=0, along any periodic solution is

M(t)=Mz(t)=cos⁡t−sin⁡t000sin⁡tcos⁡t00000cos(pt)−sin(pt)000sin(pt)cos(pt)000001.

The inverse matrix of Mt is:

MIt=cos⁡tsin⁡t000−sin⁡tcos⁡t00000cos(pt)sin(pt)000−sin(pt)cos(pt)000001.

Now computing the function F(z) given in 15, we got that the system F(z)=0, can be written as system 8 with the function Fk(X0,Y0,Z0,U0,V0) given in 6. The zeros (X0*,Y0*,Z0*,U0*,V0*) of system 8 with respect to the variables X0, Y0, Z0, U0, and V0, provide periodic orbits of system 21 with ε≠0 sufficiently small if they are simple, i.e. if 9 holds. Going back through the change of variables, for every simple zero X0*,Y0*,Z0*,U0*,V0* of system 8, we obtain a 2πn periodic solution x(t) of the differential equation (1) for ε≠0 sufficiently small such that x(t) tends to the periodic solution 10 of x(5)+(p2+1)x…+p2x⋅=0 when ε→0. Note that this solution is periodic of period 2πn. This completes the proof of Theorem 3.

5. Proof of Corollaries 2 and 4

Proof of Corollary 2.

Consider the function

F(t,x,x⋅,x¨,x…,x….)=(ax2+bx⋅)(1+cos⁡t),

which corresponds to the case p=12, λ=2. The functions Fi = Fi(X0,Y0,Z0,U0) for i=1,...,4 of Theorem 1 are:

F1=20482601aU02−10242601aU0Z0+14225aX02+16225aX0Y0+26225aY02+1282601aZ02+415bX0−215bY0,F2=215bX0+415bY0+5122601aU02−8225aX02+8225aY02−5122601aZ02−640867aU0Z0−475aX0Y0,F3=217bU0−817bZ0−64255aU0X0−3245aU0Y0−64153aX0Z0,F4=−32765aY0Z0−851bU0−251bZ0+256765aU0X0−128765aU0Y0+64765aX0Z0,

System F1=F2=F3=F4=0 has only real solution:

(X0*,Y0*,Z0*,U0*)=−6ba,3ba,0,0.

Since the Jacobian

det∂(F1,F2,F3,F4)∂(X0,Y0,Z0,U0)|(X0,Y0,Z0,U0)=(X0*,Y0*,Z0*,U0*)=208b4135≠0,b≠0

by Theorem 1 equation 1 has the periodic solution of the statement of the corollary.□

Proof of Corollary 4.

Consider the function:

F(t,x,x.,x..,x...,x....)=(2x2−x⋅2+x−2x⋅)sin⁡t,

which corresponds to the case p=12, λ=0. The functions Fi = Fi(X0,Y0,Z0,U0,V0) for i=1,...,5 of Theorem 2 are:

F1=−43X0Y0,F2=−1129U02−16V02+29X02−109Y02−1129Z02−2V0,F3=43U0−43Z0+643U0V0+169X0U0+649Y0Z0,F4=43U0+43Z0+649Y0U0+643V0Z0−169X0Z0,F5=43X0−23Y0−32U0Z0−323V0Y0.

System F1=F2=F3=F4=F5=0 has the six solutions (X0*,Y0*,Z0*,U0*,V0*) given by:

0,316,344842,0,−116, 0,316,−344842,0,−116, 0,−316,0,344842,−1160,−316,0,−344842,−116, 0,34010,0,0,−116,0,−34010,0,0,−116.

Since the Jacobian:

det∂(F1,F2,F3,F4,F5)∂(X0,Y0,Z0,U0,V0)|(X0,Y0,Z0,U0,V0)=(X0*,Y0*,Z0*,U0*,V0*),

for six solutions (X0*,Y0*,Z0*,U0*,V0*) is:

85252, 85252, −85252,−85252,−3222510,−3222510,

Respectively, we obtain using Theorem 3 the ten solutions given in statement of the corollary.□