A

The study of the existence and number of limit cycles that a differential system in ℝ^{2} can exhibit is one of the more difficult problems in the qualitative theory of the differential system in the plane. Thus in 1900, Hilbert [6] presented a list of 23 problems at the International Conference of Mathematicians in Paris; most of these problems were solved partially or completely, but the second part of the 16th problem remains unsolved till today. This problem ask about the existence of an upper bound for the maximal number of limit cycles that polynomial differential systems in ℝ^{2} of a given degree can exhibit.

A source of producing limit cycles is by perturbing the periodic orbits of a centre; see for instance the papers [3, 11] and the book of Christhoper and Li [5], and the hundreds of references quoted there.

The classical Mathieu's differential equation [9] is

Mathieu's equation can be written as the differential system

In [4], Chen and Llibre studied the limit cycles of the differential system

In the present work we study the limit cycles of the following generalisation of differential system (2) as

Our main result is as follows.

(

Theorem 1 is proved in section 3.

In section 2, we present a summary of the averaging theory of first-order and of the Descartes theorem that we shall need for proving Theorem 1.

In these subsection we summarise the result stated in Theorems 11.5 of the book of Verhulst [15] on the averaging theory. For a general introduction to the computation of periodic orbits using the averaging theory see book [8].

Consider the periodic differential system
^{1} = ℝ/(2^{1} × ^{1} × _{0}, _{0}) → ℝ are ^{2} functions, being

Now we consider the averaging function _{0}) is the solution of system (4) such that _{0}) = _{0}, then we have

In order to study the real zeros of the function

(Descartes Theorem). _{i1}^{i1} + _{i2}^{i2} + ⋯ + _{in}^{in} _{1} ≤ _{2} ≤ ⋯ ≤ _{n} and a_{ij}_{ij}a_{ij+1}_{ij} and a_{ij+1} have a variation of sign. If the number of variations of signs in the polynomial p

Let the polynomials

We write the differential system (3) in polar coordinates (

In our study we shall use the following formulas for computing the averaged function:

We consider the following three cases with

Since ^{2}, . . ., ^{n}^{+1}}. Using Descartes theorem it follows that the polynomial

Again, ^{2}, ⋯, ^{n}^{+1}}, and as in case 1 by Descartes theorem it follows that for

As in the previous two cases, we conclude that for

Working as in case 1, we compute the averaged function and we get

Since ^{3}, . . ., ^{n}^{+1}}. Using Descartes theorem it follows that the polynomial

Again working as in case 1, we compute the averaged function and we obtain

Since the polynomial ^{2}, . . . , ^{n}^{+1}}. Using Descartes theorem it follows that the polynomial

We distinguish two cases with

Now the polynomials ^{2}, . . . , ^{n}

Computing the averaged function we obtain

The polynomial ^{3}, . . . , ^{n}

Again working as in case 1, we compute the averaged function and we obtain

Since the polynomial ^{2}, ⋯, ^{n}^{+1}}. Using Descartes theorem it follows that the polynomial

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