Limit Cycles for the Class of D-Dimensional Polynomial Differential Systems

whereP(x, y) andQ(x, y) are arbitrary polynomials of degree n starting with terms of degree 2, a is a real parameter, and ε is small parameter.They proved that for ε ̸ = 0 sufficiently small, the maximum number of limit cycles bifurcating from the periodic orbits of the linear center ?̇? = −y, ?̇? = x, obtained using the averaging theory of first order, is (n − 1)/2 if n is odd and (n − 2)/2 if n is even. In the same paper, the authors studied the limit cycles of the differential system


Introduction
One of the main problems in the theory of differential systems is the study of their periodic orbits, their existence, their number, and their stability.As usual, a limit cycle of a differential system is a periodic orbit isolated in the set of all periodic orbits of the differential system.
In general, to obtain analytically periodic solutions of a differential system is a very difficult work, usually impossible.

Journal of Applied Mathematics
Here, using the averaging theory of first order, we will study the number of limit cycles of the differential system in R  , where   ( 1 , . . .,   ) for  = 1, . . .,  is a polynomial of degree  starting with terms of degree 2, ,   ∈ R, and  is a small parameter.
The problem of studying the limit cycles of system (4) is reduced using the averaging theory of first order to find the zeros of a nonlinear system of  − 2 equations with  − 2 unknowns.It is known that in general the averaging theory for finding periodic solutions does not provide all the periodic solutions of the system; this is due to two main reasons.First, the averaging theory for studying the periodic solutions of a differential system is based on the so-called displacement function, whose zeros provide periodic solutions of the differential system.This displacement function in general is not global and consequently it cannot control all the periodic solutions of the differential system, only the ones which are in its domain of definition and are hyperbolic.Second, the displacement function is expanded in power series of a small parameter , and the averaging theory only controls the zeros of the dominant term of this displacement function.When the dominant term is   , we talk about the averaging theory of order .For more details, see, for instance, [3] and the references quoted there.The averaging theory of first order necessary for the results of this paper is summarized in Section 2.
Our main result on the limit cycles of the differential system (4) is as follows.
Theorem 1 is proved in Section 3.

Limit Cycles via Averaging Theory
Roughly speaking, we can say that the averaging theory gives a quantitative relation between periodic solutions of a nonautonomous periodic differential system and the solutions of its averaged differential system, which is autonomous.The next result provides a first-order approximation in  for the limit cycles of a periodic differential system; for a proof, see Theorem 2.6.1 of [4] and Theorem 11.5 of [5].
(b) If  is a singular point of the averaged system (6) such that the determinant of the Jacobian matrix  0 /| = is not zero, then there exists a limit cycle (, ) of period  for system (5) such that (0, ) →  as  → 0.
(c) The stability or instability of the limit cycle (, ) is given by the stability or instability of the singular point  of the averaged system (6) when  is a hyperbolic singular point.
To prove Theorem 1, we need the following three lemmas which are proved in [6].
Before doing the proof of Theorem 1, we recall the Bézout theorem which will be used later on; for a proof of this result, see [7].
Thus, from Theorems 2 and 3, it follows that the maximum number of limit cycles bifurcating from the differential system (4) is  −1 obtained using the averaging theory of first order.This completes the proof of Theorem 1.