By Yen-Chi'ien Yeh

During the last twenty years the speculation of restrict cycles, particularly for quadratic differential platforms, has improved dramatically in China in addition to in different international locations. This monograph, updating the 1964 first version, comprises those contemporary advancements, as revised by means of 8 of the author's colleagues of their personal parts of workmanship. the 1st a part of the publication bargains with restrict cycles of basic airplane desk bound platforms, together with their lifestyles, nonexistence, balance, and strong point. the second one part discusses the worldwide topological constitution of restrict cycles and phase-portraits of quadratic platforms. eventually, the final part collects very important effects which may now not be incorporated lower than the subject material of the former sections or that experience seemed within the literature very lately. The booklet as a complete serves as a reference for school seniors, graduate scholars, and researchers in arithmetic and physics.

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In general, w(k)(O) can be obtained from the above method, but its representation formula becomes more complicated ask increases (see [40]). , (0) = "Ill" (0) = "Ill"' (0) = 0, etc. From the above formulas we see that in order to compute "111 11 (0), "111 111 (0), ... we first have to compute 2 (8, 0), F::~ (8, 0), ... , the coefficients of the Taylor series expansion of F(8, n) at n = 0. In the following, we introduce a method of the Japanese mathematician Minoru Urabe [41]; using this method we can also obtain, when r is a periodic cycle, the approximate representation formula for the period of a closed trajectory in the vicinity of r if we know that the period of r is T.

The original equations become dxtfdt = -a[ft(x) + h(y)} + bx1, Suppose there exists a closed trajectory equation, we obtain 0 = £[-a(ft(x) r. 34) to this + h(y)) + bx1} dy1 + x1dx1 =lis dx 1[-a(ft(x) + h(y)) + bx1} dy1 + dy 1X1 dx1 = -a IIs dx1 rp (- byl: Xl) dyt. This is a contradiction since, according to the assumption, the right side of the above formula should not be equal to zero. In addition, Krasovski1 [30] also obtained the following theorem. 15. 1) has at least one externally stable cycle and one internally stable cycle, and they may coincide to form a stable cycle.

1 and general concepts of singular points and then extended the Annular Region Theorem to regions which contain singular points and general n-multiply connected regions. Yu Shu-xiang (32] applied the results of (31] to study existence of closed trajectories of dynamical systems with singular points on the two-dimensional manifolds, and extended and improved the results of Sacker and Sell [33]; (32] also gave a new proof of • extended the theorem the theorem of (31]. In addition, Dong Zhen-xi (34] also of (31] to general two-dimensional manifolds.