Periodic Solutions of Some Polynomial Differential Systems in Dimension 3 via Averaging Theory

Abstract

We provide sufficient conditions for the existence of periodic solutions of the polynomial third order differential system ,  ,  and  , where , , and are polynomials in the variables , , and of degree with being periodic functions, is a real number, and is a small parameter.

1. Introduction and Statements of Results

One of the main problems in the theory of differential systems is the study of their periodic orbits, their existence, their number, and their stability. As usual, a limit cycle of a differential equation is a periodic orbit isolated in the set of all periodic orbits of the differential equation.

In this paper, we study the existence of the periodic orbits of the polynomial third order differential systemwhere is a real number, , , and are polynomials in the variables , , and of degree , with being periodic functions, and is a small parameter.

This problem has been studied in the homogeneous case (more precisely in the case where with ) by different authors by applying other versions of theorems of averaging method; see for instance [1–4]. More precisely, in paper [1], the authors considered the following system: where , , and are polynomials in the variables , , and of degree . They found that this system has at most limit cycles bifurcating from the periodic orbits of the linear system , , and . Moreover, there are such perturbed systems having limit cycles. In paper [2], the authors considered the following system: where , , and are arbitrary polynomials of degree starting with terms of degree 2. They found that there are systems in the previous system having at least limit cycles bifurcating from the periodic orbits of the system , , and In paper [3], the authors considered the following differential system:where , , and are polynomials in the variables , , and of degree and is a real number different from zero. They found that this system has at least limit cycles bifurcating from the periodic orbits of the linear center contained in when In [4], the authors considered the homogeneous case of system (1) with and the polynomials , , and of degrees , , and . They found that there are at most limit cycles bifurcating from the periodic orbits of the linear differential system , , and where . In [5–8], the authors studied the limit cycles of classes of three order differential equations using averaging theory. There have been many papers studying the periodic solutions to third-order differential equations using Schauder’s or Leray Schauder’s fixed point theorem (see [9–11]) or the nonlocal reduction method (see [12, 13]).

There are very few results which study the bifurcation of limit cycle from the periodic orbits for 3-dimensional systems in ; see [14, 15].

To obtain analytically periodic solutions is in general a very difficult work, usually impossible. Here with the averaging theory we reduce this difficult problem for differential system (1) to find the zeros of a nonlinear system of three equations with three unknowns. It is known that in general the averaging theory for finding periodic solutions does not provide all the periodic solutions of the system. To explain this idea, there are two main reasons. First, the averaging theory for studying the periodic solutions of a differential system is based on the so-called displacement function, whose zeros provide periodic solutions. This displacement function in general is not global and consequently it cannot control all the periodic solution, only the ones which are in its domain of definition and that are hyperbolic. Second, the displacement function is expanded in power series of a small parameter , and the averaging theory only controls the zeros of the dominant term of this displacement function. When the dominant term is , we talk about the averaging theory of order . For more details, see for instance [16] and the references quoted there. For more information about averaging theory, see Section 2 of this paper.

Our main results on the periodic solutions of the differential system (1) are the following ones.

Theorem 1. One considers differential system (1) with . One defines where Ifthen for every solution of the system satisfying differential system (1) with has a periodic solution , which tends to the periodic solution given by of the differential system when .
Note that this solution is periodic of period .

Theorem 2. One considers differential system (1) with . One defines where Ifthen for every solution of the system satisfying differential system (1) with has a periodic solution , which tends to the periodic solution given by of the differential system when .
Note that this solution is periodic of period .

Theorems 1 and 2 are proved in Section 3. Their proofs are based on the averaging theory for computing periodic orbits; see Section 2.

Corollary 3. Consider differential system (1) with where Then, this differential system has four periodic solutions with , tending to the periodic solutions where of the differential system when .
Corollary 3 is proved in Section 3.

Corollary 4. Consider differential system (1) with where Then, this differential system has two periodic solutions , tending to the periodic solutions of the differential system when .
Corollary 4 is proved in Section 3.

2. Basic Results on Averaging Theory

In this section, we present the basic results on the averaging theory that we will need for proving the main results of this paper.

We consider the problem of the bifurcation of -periodic solutions from differential systems of the formwith to being sufficiently small. Here the functions and are functions, -periodic in the first variable, and is an open subset of . The main assumption is that the unperturbed systemhas a submanifold of periodic solutions. A solution of this problem is given using the averaging theory. For a general introduction to the averaging theory, see the books of Sanders and Verhulst [17] and of Verhulst [18].

Let be the solution of system (26) such that . We write the linearization of the unperturbed system along a periodic solution as

In what follows we denote by some fundamental matrix of linear differential system (27) and by the projection of onto its first coordinates; that is, . We assume that there exists a -dimensional submanifold of filled with -periodic solutions of (26). Then, an answer to the problem of bifurcation of -periodic solutions from the periodic solutions contained in for system (25) is given in the following result.

Theorem 5. Let be an open and bounded subset of , and let be a function. We assume that(i) and that for each the solution of (26) is -periodic;(ii)for each there is a fundamental matrix of (27) such that the matrix has in the upper right corner the zero matrix and in the lower right corner a matrix with .
We consider the function :If there exists with and , then there is a -periodic solution of system (25) such that as .
Theorem 5 goes back to Malkin [19] and Roseau [20]; for a shorter proof see [21].

We assume that there exists an open set with such that, for each , is -periodic, where denotes the solution of the unperturbed system (26) with . The set is isochronous for system (25); that is, 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 -periodic solutions from the periodic solutions contained in is given in the following result.

Theorem 6 (perturbations of an isochronous set). One assumes that there exists an open and bounded set with such that, for each , the solution is -periodic, considering a function defined byIf there exists an with and , then there exists a -periodic solution to system (25) such that as .
For the proof of Theorem 6, please see Corollary of [21].

3. Proof of Theorems and Corollaries

The solution of system (1) with such that iswhere

So

For studying the periodicity of this solution, we distinguish the two cases: and . These two cases will be studied, respectively, in Theorems 1 and 2.

3.1. Proof of Theorem 1

We will apply Theorem 6 to differential system (1) with . We note that system (1) can be written as system (25) taking

We will study the periodic solutions of system (26) in our case, that is, the periodic solutions of system (1) with and . The solutions (32) with such that are

These solutions are -periodic if and only if

We obtain the following periodicity conditions:

The set of periodic solutions (34) has dimension 3. To look for the periodic solutions of our system (1) with , we must calculate the zeros of the system , where is given by (29). The fundamental matrix of the differential system (27) is

Now computing the function given in (29), we find that the system can be written aswhere where , , and have been defined in the statement of Theorem 1. The zeros of the systemwith respect to the variables , , and , provide periodic solutions of system (1) with and being sufficiently small if they are simple, that is, if

For simple zeros of system (40), we obtain a -periodic solution of differential system (1) with , for being sufficiently small which tends to the periodic solution of the differential system when .

This completes the proof of Theorem 1.

3.2. Proof of Theorem 2

We will apply Theorem 5 to differential system (1) with . It can be written as system (25) taking

We will study the periodic solutions of system (26) in our case, that is, the periodic solutions of system (1) with and . The solution (32), with such that , is

These solutions are -periodic if and only if

We obtain the following periodicity conditions:

The set of the periodic solutions becomes

The set of the periodic solutions (48) has dimension two. To look for the periodic solutions of our system (1) with , we must calculate the zeros of the system , where is given by (28). The fundamental matrix of the differential system (27) is

It verifies

Consequently, all the assumptions of Theorem 5 are satisfied. Therefore, we must study the zeros of the system of two equations with two unknowns, where is given in the statement of Theorem 5. More precisely, we have where where , , and have been defined in the statement of Theorem 2. The zeros of the systemwith respect to the variables and , provide periodic solutions of system (1) with and being sufficiently small if they are simple, that is, if

For simple zeros of system (52), we obtain a -periodic solution of differential system (1) with , for being sufficiently small which tends to the periodic solution of the differential system when .

This completes the proof of Theorem 2.

3.3. Proof of Corollary 3

We must apply Theorem 1 with

First, we can verify easily conditions (7)

Computing the functions , , and of Theorem 1, we obtain

The system has four solutions given by , , , and .

Since for these four solutions is , , , and , respectively, we obtain using Theorem 1 the solutions given in the statement of Corollary 3.

3.4. Proof of Corollary 4

We must apply Theorem 2 with

First, we can verify easily conditions (14):

Computing functions and of Theorem 2, we obtain

The system has two solutions given by , .

Since for these two solutions is and respectively, we obtain using Theorem 2 the solutions given in the statement of Corollary 4.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.