Limit Cycles for the Class of -Dimensional Polynomial Differential Systems

Abstract

We perturb the differential system , , and for inside the class of all polynomial differential systems of degree in , and we prove that at most limit cycles can be obtained for the perturbed system using the first-order averaging theory.

1. Introduction

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 system is a periodic orbit isolated in the set of all periodic orbits of the differential system.

In [1], the authors studied the differential systemwhere and are arbitrary polynomials of degree starting with terms of degree , is a real parameter, and is small parameter. They proved that for sufficiently small, the maximum number of limit cycles bifurcating from the periodic orbits of the linear center , , obtained using the averaging theory of first order, is if is odd and if is even. In the same paper, the authors studied the limit cycles of the differential systemwhere , , and are arbitrary polynomials of degree starting with terms of degree and . Then, there exists sufficiently small such that for there are systems (2) having at least limit cycles bifurcating from the periodic orbits of the system , , and .

In [2], the authors studied the number of limit cycles of the differential systemwhere , , and are polynomials of degree starting from terms of degree . Then, there exists sufficiently small such that for there are systems (3) having at least limit cycles bifurcating from the periodic orbits of the system , , and .

In general, to obtain analytically periodic solutions of a differential system is a very difficult work, usually impossible. Here, using the averaging theory of first order, we will study the number of limit cycles of the differential systemin , where for is a polynomial of degree starting with terms of degree , , and is a small parameter.

The problem of studying the limit cycles of system (4) is reduced using the averaging theory of first order to find the zeros of a nonlinear system of equations with unknowns. It is known that in general the averaging theory for finding periodic solutions does not provide all the periodic solutions of the system; this is due to 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 of the differential system. This displacement function in general is not global and consequently it cannot control all the periodic solutions of the differential system, only the ones which are in its domain of definition and 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, [3] and the references quoted there. The averaging theory of first order necessary for the results of this paper is summarized in Section 2.

Our main result on the limit cycles of the differential system (4) is as follows.

Theorem 1. By applying the first-order averaging theory to the polynomial differential system (4), for sufficiently small at most limit cycles bifurcate from the periodic orbits of the differential system , , , and .

Theorem 1 is proved in Section 3.

2. Limit Cycles via Averaging Theory

Roughly speaking, we can say that the averaging theory gives a quantitative relation between periodic solutions of a nonautonomous periodic differential system and the solutions of its averaged differential system, which is autonomous. The next result provides a first-order approximation in for the limit cycles of a periodic differential system; for a proof, see Theorem of [4] and Theorem of [5].

Theorem 2. One considers the following two initial-value problems:where , , , and is an open subset of , , , , , and are periodic of period in the variable , and is the averaged function of with respect to ; that is, Assume that(i), its Jacobian , its Hessian , , and its Jacobian are defined, continuous, and bounded by a constant independent of in and ;(ii) is a constant independent of ;(iii) belongs to on the time interval Then, the following statements hold:(a)On the time scale , we have that as (b)If is a singular point of the averaged system (6) such that the determinant of the Jacobian matrix is not zero, then there exists a limit cycle of period for system (5) such that as .(c)The stability or instability of the limit cycle is given by the stability or instability of the singular point of the averaged system (6) when is a hyperbolic singular point.

To prove Theorem 1, we need the following three lemmas which are proved in [6].

Before doing the proof of Theorem 1, we recall the Bézout theorem which will be used later on; for a proof of this result, see [7].

Theorem 3 (Bézout theorem). Let be polynomials in the variables of degree for Consider the following polynomial system: , where . If the number of solutions of this system is finite, then it is bounded by .

Lemma 4. For , one defines Then, if and only if is even. For with even, one has

Lemma 5. The following equalities hold. For , one has

Lemma 6. For , one has

3. Proof of Theorem 1

Doing the change to polar coordinates , , system (4) becomes where . Taking as the new independent variable instead of , this differential system can be written as for , where with Now, using the notation introduced in Lemma 4 and applying the first-order averaging method, we must find the zeros of the system where for , and

Theorem 7. Let and . The function is a polynomial of degree in the variables and , while is a polynomial of degree . Moreover, , where is a polynomial in the variables and of the degree at most .

Proof. The function is a linear combination of and , where .
Lemma 4 claims that where is an even polynomial of the degree if is odd and of degree otherwise. Using the variable , we conclude that where is an odd polynomial of degree or . Since vanishes at , the functions for span the space of functions of the form vanishing at with or , respectively. Lemma 4 implies that any function is of the form where is an even polynomial in of the degree ( is necessarily even by Lemma 4) and is a polynomial in of the degree or . We conclude that the functions , where , generate the space of functions , where (and, in addition, ). Therefore, ,
In a similar way, is a linear combination of and terms and , where . We conclude that the functions and , where , generate the space of functions , where . We have which implies . Therefore, . So the polynomials and in the variables have at most degree . Hence, by the Bézout theorem, the maximum number of solutions of for and is at most for .

Thus, from Theorems 2 and 3, it follows that the maximum number of limit cycles bifurcating from the differential system (4) is obtained using the averaging theory of first order. This completes the proof of Theorem 1.

4. An Application of Theorem 1

In system (4), we consider the case even and Computing the averaged functions and taking , we have where is an arbitrary polynomial in of degree such that . At the same time, the averaged function corresponding to satisfies for . It is easy to obtain the following relations: Looking at the second term of the first relation and at the first term of the second relation, we obtain that and for even are independent. In particular, using Lemmas 4, 5, and 6, we obtain where where Writing , the polynomials satisfy the conditions for and . Then, we can define a polynomial of degree in : Due to the independence of and and the arbitrariness of the coefficients and , the polynomial is an arbitrary polynomial such that . In fact, it is obvious that and have parameters, respectively, where coefficients allow choosing the first term of arbitrarily except for the term with , implying that the even terms of are arbitrary except for the constant term, while the other coefficients allow choosing the second term in arbitrarily, implying that the odd terms of are arbitrary. Therefore, the polynomial of the degree satisfies and has arbitrary coefficients. The number of solutions of , , for , is equal to the number of the intersection points of the curvesHence, by the Bézout theorem, the maximum number of the common solutions of system (30) is at most for . We can find intersection points on with for , , , which (using the averaging theory; see Theorem 2) give rise to limit cycles bifurcating from periodic orbits of the system , , , and .

Conflict of Interests

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