RESEARCH ON THE COMPOSITION CENTER OF A CLASS OF RIGID DIFFERENTIAL SYSTEMS

In this paper, we answer the question: under what conditions a class of rigid differential systems have a composition center. We give the sufficient and necessary conditions for these systems to have a center at origin point. At the same time, we give the formula of focal values and the highest order of fine focus.

This system is called a rigid system [3] because the derivative of the angular variable is constant. It is clear that the origin is the only critical point and if it is a center then it is a uniformly isochronous center [12]. In [9,16], the authors have proved that a planar polynomial differential system of degree n+1 has a center at the origin of coordinates, then this center is uniform isochronous if and only if by doing a linear change of variables and a scaling of time it can be written as (1.1). The interest in the uniform isochronous centers has attracted people's attention since the 17th century. So far, there are many people who have strong interest in this problem and have achieved fruitful results [2,8,9,12,16]. In [1,2] the authors have used techniques based on normal forms and commutation and have proved that the rigid system (1.1), in the cases: P = P 1 + P n or P = P 2 + P 2m or P = P 1 + P 2 + P 3 + P 4 , it has a center if and only if it is reversible. In [20], the author calculated by computer to get the center condition for this system with P = P k + P 2k (k = 2, 3,4,5). In [14,15], the numbers of limit cycles of (1.1) have been discussed. In [18,19], some new methods have been used to studied the center problem of this system. In this paper, we consider the following rigid system    x = −y + x(P 1 (x, y) + P 3 (x, y) + P 7 (x, y)), y = x + y(P 1 (x, y) + P 3 (x, y) + P 7 (x, y)). (1.2) By [5,6], this system has a center at (0, 0) if and only if all solutions r(θ) of periodic differential equation dr dθ = r(P 1 (cos θ, sin θ)r + P 3 (cos θ, sin θ)r 3 + P 7 (cos θ, sin θ)r 7 ) (1. 3) near the solution r = 0 are periodic, r(0) = r(2π). In such case it is said that equation (1.3) has a center at r = 0. As we known, the derivation of conditions for a center is a difficult and longstanding problem in the theory of nonlinear differential equations, however due to complexity of the problem necessary and sufficient conditions are known only for a very few families of polynomial systems [4,13,17]. In [5,6] the authors introduce a simple condition called Composition Conditions, which ensures that the Abel equation has a center. Roughly speaking the composition condition says that the primitives of the functions A and B depend functionally on a new 2π-periodic function. When an Abel equation has a center because A and B satisfy the composition condition we will say that the equation has a Composition Center [6]. The Composition Conjecture is that the composition condition is not only the sufficient but also necessary condition for a center. This conjecture first appeared in [7] with classes of coefficients which are polynomial functions in t, or trigonometric polynomials. A counterexample was presented in [5] to demonstrate that the conjecture is not true. To find the restrictive conditions under which the composition conjecture is true, this is an open problem which has attracted during the last years a wide interest. In [5] the author has proved that for a family of cubic system the composition conjecture is valid. [21,22] the author used the different method from [1,2] to prove that for system (1.1) with P = P 1 + P n and P = P 2 + P 2m , the composition conjecture is true. The authors in paper [6,10,11] give the sufficient and necessary conditions for the r = 0 of the Able equation (1.4) to be a composition center.
In this paper, we find out all the restrictive conditions under which the origin point of (1.3) is a composition center. At the same time, we give the sufficient and necessary conditions for equation (1.2) to have a center at origin point by using a different method from [1,2,20]. These center conditions are more succinct and beautiful than those calculated by computer.
The condition in Lemma2.1 is called the Composition Condition. This is a sufficient but not a necessary condition for r = 0 to be a center [7,10].
The following statement presents a generalization of Lemma 2.1.

Lemma 2.2 ( [23]
). If there exists a differentiable function u of period 2π such that for some continuous functionsÂ i (i = 1, 2, ..., n), then the differential equation has a center at r = 0.
Denote : , U = a cos θ + b sin θ, V = a sin θ − b cos θ, then P 1 = p 2 10 + p 2 01 U,P 1 = p 2 10 + p 2 01 V + p 01 , Obviously, P 3 and P 7 can be rewritten in the following forms where n i (i = 1, 3), s j , t j (j = 1, 3, 5, 7) are real numbers and i.e., solving these equations we get n 1 = n 3 = 0, so, since the value of the determinant of the coefficient matrix of the above equations is not equal to zero, so t 1 = t 3 = t 5 = t 7 = 0 and
Example 3.1. The system