The study of the existence and number of limit cycles that a differential system in ℝ2 can exhibit is one of the more difficult problems in the qualitative theory of the differential system in the plane. Thus in 1900, Hilbert  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 ℝ2 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 , and the hundreds of references quoted there.
The classical Mathieu's differential equation  is
Mathieu's equation can be written as the differential system
In , 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.
In these subsection we summarise the result stated in Theorems 11.5 of the book of Verhulst  on the averaging theory. For a general introduction to the computation of periodic orbits using the averaging theory see book .
Consider the periodic differential system
Now we consider the averaging function
In order to study the real zeros of the function
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
As in the previous two cases, we conclude that for
Working as in case 1, we compute the averaged function and we get
Again working as in case 1, we compute the averaged function and we obtain
Since the polynomial
We distinguish two cases with
Now the polynomials
Computing the averaged function we obtain
Again working as in case 1, we compute the averaged function and we obtain
Since the polynomial
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