【摘要】 <正> 我们考虑常微自治系统其中β是大于零的常数,F_x,y与G_i（x,y）是x与y的i次齐次式。依照Andronov的理论（[1]中§25）,若O（o,o）是（E_n）的k重焦点,则系统（E_n）的2k+1级近似系统在O（o,o）附近最多有k个极限环。但是要实际判定任意一个这样的系统确有k个极限环更多还原

【Abstract】 In this paper, the following dynamic systemwas considered where β = const. >0, Fi and Gi are homogeneous polynomials of order i. It is known that if O (o, o) is a multiple focus of multiplicity k of the system (En), then a system which sufficiently closes to (En) to rank 2k+1 has at most k limit cycles in a neighborhood of the focus. The main purpose of the present paper is to give a sufficient condition to have precisely k limit cycles about a multiple focus of multiplicity k.THEOREM 1 Let O(o, o) be a multiple focus of multiplicity k (k>1) of the system (En) and d(2k+1-27) (o, δ1,..., δ1) the first nonzero focal values of the systemwhereand g1 (1≤l≤k-1) are polynomials of x and y of ordern, and fk and gk are linear functions of x and y. Iffor all l. 1≤l≤k-1, and all δj-belonging a neighborhood of δ = 0, j = l,..., k, then for sufficiently small parameters δ1,...,δk the dynamic systemhas k limit cycles in a neighborhood of O(o, o). In this paper we provedwhere P2m is the first nonzero term in the sequence introduced by F. Gobber and K. D. Willamowski [3]. When 2≤n≤7, and m≤4, P2m can be expressed by the coefficients of (En). Therefore one may use theorem 1 directly in terms of the coefficients of approximate system of (En).Applying Theorem 1 to the system (E2), we obtained a simple proof of the part results of [4], [6] and [7].更多还原