Periodic solutions for a class of fifth-order differential equations Open Access

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. [1–6]. 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.

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

$satisfying$

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

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 is proved in Section 5.

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:

$satisfying$

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

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 is proved in Section 5.

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:

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:

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:

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.

We consider the function $F:CL(W)→Rk$

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.

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].

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

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:

where

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

where

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

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:

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:

The inverse matrix of $Mt$ is:

Moreover, an easy computation shows that:

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.

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

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:

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:

where

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.:

where

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

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:

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