SOME PROPERTIES OF THE DULAC FUNCTIONS SET

In order to rule out the existence of periodic orbits in the plane for a given system of differential equations, we discuss the feature of the set of Dulac functions, establishing some of its properties as well as some results for special cases where this set of functions is not empty. We give some examples to illustrate applications of these results.


Introduction
Many problems of the qualitative theory of differential equations in the plane refer to the existence of periodic orbits, for example in mechanical or electrical engineering, biological models and many others.However, until now we can not answer in general whether, given an arbitrary system of differential equations, it has periodic orbits or not.
There are some criteria that allow us to rule out the existence of periodic orbits in the plane such as Poincaré-Bendixson, the index theory and special systems such as the system gradient, among others, see ( [1], [9], [8] and [5]).
A classical criterion to discard the existence of periodic orbits (or limiting the number of these) in a given region is the Bendixson-Dulac theorem.
does not change sign in D and vanishes at most on a set of measure zero.Then the system does not have periodic orbits in D. EJQTDE, 2011 No. 72, p. 1 According to this criterion, to rule out the existence of periodic orbits of the system (1) in a simply connected region D, we need to find a function h(x 1 , x 2 ) that satisfies the conditions of the theorem of Bendixson-Dulac.Such function h is called a Dulac function.
Usually it is not easy to determine such a function, however it is possible to propose some candidates of the form h = 1, x s 1 , x s 2 , e ax 1 +bx 2 , x s 1 x t 2 , s, t ∈ Q, a, b ∈ R, among others.In the particular case h = 1 this theorem is known as Bendixson's criterion.
In this paper we will introduce and study the set H + D (F ) of Dulac functions for a region D and the vector field F = (f 1 , f 2 ) defined by system (1), showing some characteristics that allow us to say whether the set H + D (F ) is different from the empty set.

Properties of the Dulac functions
Consider the vector field now let C 0 (D, R) be the set of continuous functions and define the set Also for the simply connected region D, we introduce the sets A Dulac function in the system (1) of the Bendixson-Dulac theorem is an element in the set . This set has the following properties that are listed below in the next result.
(f) Suppose there is a periodic orbit γ in D. Take D 1 as the region bounded by γ, then by hypothesis, there exists a function h ∈ H + D 1 (F ) and so, D 1 can not have periodic orbits. 2 Now we examine conditions that imply that the set H + D (F ) = ∅.Our results are established with the help of the techniques developed by the authors in [10], let us recall the following proposition Theorem 2. ( [10]).If there exist c ∈ F D such that h is a solution of the equation A first result of the existence of Dulac functions is as follows Theorem 3. Suppose there is c ∈ F D , such that and is continuous, then the set H + D (F ) is not empty.EJQTDE, 2011 No. 72, p. 3 Proof.We consider the case µ 1 depending only on x 1 .We seek a Dulac function, using the theorem 2, so that the associated equation is Assume that h depends only on x 1 .Thus the previous equation reduces to which is rewritten as From our hypothesis h = exp µ 1 dx 1 is a solution and satisfies the conditions of theorem 2, therefore the system has a Dulac function.The proof is complete.2 Example 1.Consider the system or replacing We have µ 1 = 2x 1 cos x 1 and therefore the set We can take c ∈ F R 2 as c(x 1 , x 2 ) := 6x 2 1 ≥ 0, we have , that only depend on x 2 , then by theorem 3, Now we use theorem 3 to study some special systems, consider an equation as follows We establish the following From theorem 3, it is enough to see that we can choose µ 1 (x 1 ) a continuous function such that Without loss of generality, suppose r 2 ≥ 0 and r 1 > 0 in D. We take µ 1 (x 1 ) := nr 1 (x 1 ) with n ∈ N such that µ 1 r 1 + r ′ 1 > 0 in D 0 .This is possible because r ′ 1 is continuous and D 0 compact.So we have r 2 (µ , and consider the system Another result that helps us to establish conditions for which the set suppose that there exists a function h : D → R, C 1 which only vanishes on a set of measure zero such that then for any D 1 ⊂ D simply connected compact, we have and take m 0 > 0 such that |h R) : f doesn't change sign and vanishes only on a measure zero set}.
then there are no periodic orbits in D. Proof.items (a), (b) and (d) are direct from the definition.(c) It follows from (b).