On the Limit Cycles of a Class of Generalized Kukles Polynomial Differential Systems via Averaging Theory

Abstract

We apply the averaging theory of first and second order to a class of generalized Kukles polynomial differential systems to study the maximum number of limit cycles of these systems.

1. Introduction

One of the main topics in the theory of ordinary differential equations is the study of limit cycles: their existence, their number, and their stability. A limit cycle of a differential equation is an isolated periodic orbit in the set of all periodic orbits of the differential equation. The second part of the 16th Hilbert’s problem [1] is related to the least upper bound on the number of limit cycles of polynomial vector fields having a fixed degree. This problem and the Riemann conjecture are the only two problems on the list of Hilbert which have not been solved. Here we consider a very particular case of the sixteenth Hilbert problem. We study the upper bound of the generalized Kukles polynomial system where is a polynomial with real coefficients of degree .

Kukles [2], in 1944, introduced the differential system and he gives the necessary and sufficient conditions in order that this system has a center at the origin. This cubic system without the term was also studied in [3] and the authors called it reduced. In [4] a description of the bifurcation of its critical period appears, and [5] presents the existence of reduced Kukles systems with five limit cycles. In the paper [6], the author studied the class of reduced Kukles systems under the cubic perturbation where is small and is a constant.

In [7] the author proves that some cubic systems of form (1) can have seven limit cycles. In [8], Chavarriga et al. studied the maximum number of small amplitude limit cycles for Kukles systems which can coexist with some invariant algebraic curves. Also they give a family of cubic Kukles systems with an invariant hyperbola with , which coexist with one or two small amplitude limit cycles. In [9] the author studied the maximum number of limit cycles of the generalized polynomial Liénard differential equations by using the first and second averaging method. In [10], Llibre and Mereu studied the maximum number of limit cycles of the Kukles polynomial differential systems where for every the polynomials ,  , and have degrees ,  , and , respectively, is a real number, and is a small parameter.

In this work we study the maximum number of limit cycles given by averaging theory of first and second order, which can bifurcate from the periodic orbits of the linear center ,   perturbed inside the following class of generalized Kukles polynomial differential systems: where for every the polynomials ,  , and have degrees ,  , and , respectively, and is a small parameter. We have considered the same polynomial as the coefficient of and . With this choice, we can apply the first and second order of the averaging method. If we consider the coefficients of for as arbitrary polynomials, it is difficult to apply the second order averaging method, because to pass from the first order to the second order averaging method, we must put the averaged function of the first order (see (8)) identically null. In this case the calculations of the averaged function of the second order (see (9)) become difficult. If we replace by in the coefficient of of the differential systems (5), we can apply the first order averaging method but it is not easy to apply the second averaging method. To apply the second averaging method, we must put which is equivalent to cancel all coefficients of the polynomial . The conditions on the coefficient of make the calculations of difficult. We have used the averaging method for looking for the limit cycles of many classes of Liénard systems. Here we do the same for Kukles differential systems. Comparatively, with the results of the paper [10], we obtained more limit cycles than the results of this paper. More precisely our main result is the following.

Theorem 1. Assume that for the polynomials ,  , and have degrees ,  , and , respectively, with . Then for sufficiently small the maximum number of limit cycles of the Kukles polynomial differential systems (6) bifurcating from the periodic orbits of the linear centre ,  , using averaging theory (a)of first order is(i)no limit cycle for ,(ii) limit cycles for ,(b)of second order is ,
where denotes the integer part function.

2. First and Second Order Averaging Method

In proof of our main result we use the averaging theory as it is presented in [11]. Consider the differential system where ,   are continuous functions, -periodic in the first variable, and is an open subset of . Assume that the following hypotheses (i) and (ii) hold.

(i) for all , , , , and are locally Lipschitz with respect to , and is differentiable with respect to . We define

(ii) For an open and bounded set and for each , there exists such that and .

Then, for sufficiently small there exists -periodic solution of system (7) such that .

The expression means that the Brouwer degree of the function at the fixed point is not zero. A sufficient condition for the inequality to be true is that the Jacobian of the function at is not zero.

If is not identically zero, then the zeros of become mainly the zeros of for sufficiently small. In this case the previous result provides the averaging theory of first order.

If is identically zero and is not identically zero, then the zeros of are mainly the zeros of for sufficiently small. In this case the previous result provides the averaging theory of second order. For more information about the averaging theory see [12, 13].

3. Proof of Theorem 1

3.1. Proof of Statement (a) of Theorem 1

In order to apply the first order averaging method we write system (6) with , in polar coordinates where , , . If we take , , and , system (6) can be written as follows: where If we take as a new independent variable, system (10) becomes By using the notation introduced in Section 2 we have that We know that

Let be a positive integer. We define as the largest even integer less than or equal to , and as the largest odd integer less than or equal to .

Hence We obtain

For the case , we obtain that

There is no positive root for .

For , the polynomial has at most positive roots. Hence (a) of Theorem 1 is proved.

3.2. Proof of Statement (b) of Theorem 1

For proving statement (b) of Theorem 1 we will use the second-order averaging theory. If we write

then system (6) with in polar coordinates , , becomes where

Taking as the new independent variable system, (19) can be written as where and .

In order to apply the averaging theory of second order, must be identically zero. Therefore from (16), is identically zero if and only if for even.

Now we determine the corresponding function

For this we compute where , , , and are constants.

The integral will be given in several lemmas.

Lemma 2. The integral is, in the variable , the polynomial where and are real constants.

Proof. We have that(a₁) (b₁)  where for even and odd,(c₁) (d₁) (e₁)  where for even and odd.
We have that the sum of the integrals (a₁)–(e₁) is polynomial (3.6). This ends the proof of the lemma.

Lemma 3. The integral is, in the variable , the polynomial where and are real constants.

Proof. We have that(a₂)  where for odd and even.(b₂) (c₂)  where for odd and even,(d₂) (e₂)
We have that the sum of the integrals (a₂)–(e₂) is polynomial (32). This ends the proof of the lemma.

Lemma 4. The integral is, in the variable , the polynomial where and are real constants.

Proof. We have that(a₃) (b₃)  where for even and odd,(c₃) (d₃) (e₃)  where for even and odd.
We have that the sum of the integrals (a₃)–(e₃) is polynomial (39). This completes the proof of the lemma.

Lemma 5. The integral is, in the variable , the polynomial where and are real constants.

Proof. We have that(a₄)  where for odd and even,(b₄) (c₄)  where for odd and even,(d₄) (e₄)
We have that the sum of the integrals (a₄)–(e₄) is polynomial (46). This ends the proof of the lemma.

From Lemmas 2–5 we have that Now we calculate the integral .

First we calculate . Noting that is given by (20), we have

Hence Now noting that is given by (11), we compute Hence

From (55) and (57) we obtain

Then is the polynomial

Note that in order to find the positive roots of after dividing by , we must find the zeros of a polynomial in the variable of degree equal to the max, we conclude that has at most positive roots. Hence (b) of Theorem 1 is proved.

Conflict of Interests

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