Maximum Number of Limit Cycles of a Second Order Diﬀerential System

: In this paper, we study the limit cycles of a perturbed diﬀerential system in R 2 , given by (cid:26) . x = y, . y = − x − ǫ (1 + sin n ( θ )cos m ( θ )) H ( x,y ) , where ǫ > 0 is a small parameter, m and n are non-negative integers, tan( θ ) = y/x , and H ( x,y ) is a real polynomial of degree l ≥ 1. Using Averaging theory of ﬁrst order we provide an upper bound for the maximum number of limit cycles. Also, we provide some examples to conﬁrm and illustrate our results.


Introduction
The limit cycle is one of the most used notion in the qualitative theory of planar differential system recently with the arising of its maximum number determination problem.The study of limit cycles was initiated by Poincaré [8] in 1881 where he defined this notion as a periodic orbit isolated in the set of all periodic orbits of a differential system.He defined also the notion of a center of a real planar differential system which it is an isolated critical point in the neighborhood of periodic orbits.In this paper, we will produce limit cycles by perturbing the periodic orbits of a center using Averaging theory, which it is a powerful tool for studying the existence, number and behavior of limit cycles.There are many results concerning the maximum number of limit cycles bifurcating from the linear center of planar polynomial differential systems using averaging theory (see for instance [5], [6], [2] ).
x + b(1 + cos t)x = 0, (1.1) where b is a real constant, is the simplest mathematical model of an excited system depending on a parameter, it describes the dynamics of a system with harmonic parametric excitation and a nonlinear term corresponding to a restoring force.In [4] T. Chen & J. Llibre studied the limit cycles of a kind of generalization of the equation (1.1).They considered the second order differential system where ǫ > 0 sufficiently small, m is an arbitrary non negative integer, Q(x, y) is a polynomial of degree n ≥ 1 and θ = arctan(y/x).They studied the maximum number of limit cycles which can bifurcate from 2 A. Brik and A. Boulfoul the linear center ẋ = y, ẏ = −x of the previous system with ǫ = 0.They obtained an upper bound for this system with respect to the parity of m and n.In this work, using the averaging theory of first order, we study the maximum number of limit cycles of the following differential system .

Preliminaries
In this section, we give some tools that we shall use for proving the main result.The first tool is the averaging averaging which is the basic method that we will use in our work.Consider the differential system at the following initial value where the averaged function is defined by The following theorem gives us the conditions for which the singular points of the averaged system (2.2) provide T − periodic orbits of the system (2.1).For a proof for this theorem, see ( [3], [9]).
Theorem 2.1.Consider the system (2.1) and suppose that the vector functions F, R, D x F, D 2 x F and D x R are continuous and bounded by a constant M (independent of ε) in [0, ∞) × D with −ε 0 < ε < ε 0 .Then In addition, suppose that F and R are T -periodic in t with T independent of ε.
2. If the singular point y = p of the averaged system (2.2) is hyperbolic then for |ε| > 0 small enough, the corresponding periodic solution of the system (2.1) is unique, hyperbolic and has the same kind of stability of p.
In order to study the simple zeros of the averaged function we shall apply the Descartes Theorem.

Theorem 2.2 (Descartes Theorem).
Let be a polynomial with real coefficients, with 0 ≤ i 1 < i 2 < ... < i n and a ij = 0 real constants for j ∈ 1, 2, ..., n.When a ij a ij+1 < 0, we say that a ij and a ij+1 have a variation of sign.If the number of variations of the signs is m, then p(r) has at most m positive real zeros.Furthermore, we can choose the coefficients of p(r) in such a way that p(r) has exactly n − 1 positive real zeros.

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For a proof of the previous Theorem see [1].
Before we get to compute the averaged function, we introduce some of the needed formulas For more information on these integrals see [10].

Main Result
Our main result is the following one.
Theorem 3.1.Using the averaging theory of first order, the maximum number of limit cycles of the differential system (1.2) bifurcating from the periodic orbits of the linear center ẋ = y , ẏ = −x is as follows 1.If l is even (1.a)we have at most l − 1 limit cycles, if m is even and n is odd or m is odd and n is even.
(1.b) we have at most (l − 2)/2 limit cycles, if m is even and n is even or m is odd and n is odd.

If l is odd
(2.a) we have at most l limit cycles, if m is even and n is odd or m is odd and n is even.
(2.b) we have at most (l − 1)/2 limit cycles, if m is even and n is even or m is odd and n is odd.
Theorem 3.1 is proved in the following subsection.

Proof of Theorem 3.1
Assume that the polynomial H(x, y) = l i+j=0 a ij x i y j .By applying change of the variables (x, y) into the polar coordinates (r, θ) defined by x = r cos(θ), y = r sin(θ), with r > 0, the system (1.2) becomes From the previous differential system we take θ as the new independent variable as follows Now, we compute the averaged function f , associated with the previous equation, which is given by We distinguish two cases for the parity of l, each case has four subcases for the parity of m and n.
Case (1) .Suppose that l is even, we have two subcases for studying f (r).
Subcase (1.1)If m is even and n is odd, we have Subcase (1.2) If m and n are even, we have Subcase (1.3)If m and n are odd, we have

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A. Brik and A. Boulfoul Subcase (1.4)If m is odd and n is even, we have

Case (2)
. If l is odd, then we have 4 subcases.Subcase (2.1)If m is even and n is odd, we have Subcase (2.2) If m and n are even, we have Subcase (2.3)If m and n are odd, we have Subcase (2.4)If m is odd and n is even, we have Finally, we shall discuss the number of zeros of the averaged function f (r) in the cases above using the Descartes Theorem.
From the subcases (1.1), (1.4), (2.1) and (2.4) we obtain that the function f (r) is generated by a linear combination of a set ζ 1 = {r, r 2 , ..., r p } with p ∈ {l, l + 1}.Using Descartes Theorem, it results that f (r) can have at most l − 1 solutions if l is even, m and n do not have the same parity.Also it can have l solutions if l is odd, m and n have not the same parity.Consequently, by Theorem 2.1, for ǫ > 0 small enough, the differential system (1.2) can have at most l − 1 or l limit cycles.From the subcases (1.2), (1.3), (2.2) and (2.3) we obtain that the averaged function f (r) is generated by a linear combination of a set ζ 2 = {r, r 3 , ..., r p } with p ∈ {l − 1, l}.Using Descartes Theorem, it results that f (r) can have at most (l − 2)/2 solutions if l is even, and, m and n have the same parity.Also it can have (l − 1)/2 solutions if l is odd, and, m and n have the same parity.Similarly, by using the Theorem 2.1 and for ǫ > 0 small enough, the system (1.2) can have at most (l − 2)/2 or (l − 1)/2 limit cycles.

Applications
In this section, we provide some numerical examples.
We take θ as an independent time variable, we find where Calculating the averaged function f (r), we obtain We take θ as an independent time variable, we obtain  Consequently, since n = 1, m = 1 and l = 3, from Theorem 3.1, the system (4.4) can have at most one limit cycle.