By Yen-Chi'ien Yeh

During the last 20 years the speculation of restrict cycles, specifically for quadratic differential structures, has advanced dramatically in China in addition to in different nations. 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 offers with restrict cycles of normal aircraft desk bound structures, together with their lifestyles, nonexistence, balance, and specialty. 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 that can no longer be incorporated less than the subject material of the former sections or that experience seemed within the literature very lately. The publication as a complete serves as a reference for school seniors, graduate scholars, and researchers in arithmetic and physics.

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**Sample text**

0, and (u 1 (t}, v1 (t)) is a solution which is linearly independent of (r,o'{t), 1/J'(t)). Let Z(t) = eTB x- 1 (t). It is easy to see that Z{t) = Z(t + T), where (3 and z-•(t) = X(t}e-TB = -ht/T ( Ute' Vte ht/T' (3r,o'(t)) (31/J'(t) , IZ- 1 (t)l-# o. 33) §2 . MULTIPLICITY AND STABILITY 37 which is periodic with respect to s. ~ = cp'(s) + [~e-hs/T u1(s) + ! (e-ha/T u1 (s))] z + R1 (s, z), where R1 ( s, z) has period T with respect to s, and is an infinitesimal of second order with respect to z.

EXAMPLE 5. , they must have a point of discontinuity in any simply connected region containing x 2 + y 2 = 1. 10, since we want to apply Green's formula, we have to assume P(x, y) and Q(x, y) have continuous partial derivatives in the region G. N. P. 34) where the double integral on the right is a Stieltjes integral, it could be understood as the limit of L[dxP(x, y)dy + dyQ(x, y)dx] and the requirement on P and Q could be kept the same without strengthening it. N. N. Krasovksil [29] had made use of the above rewritten Green's formula to study the global stability of the zero solution of some nonlinear secondorder differential equations.

Translator's note: A single or coarse cycle is referred to in English journals as hyperbolic. §2 . MULTIPLICITY AND STABILITY 31 Using a well-known theorem in differential calculus, we can in general establish THEOREM 2. 5. If, for a given closed trajectory w'(O) = w"(O) = ... 20) when k is odd, then r is a stable (unstable) limit cycle. If w'(o) = w"(o) = ... 21) is a semistable limit cycle. REMARK. 21) is also a necessary condition for being stable (unstable) or semistable. 3. 21) is called a k- fold limit cycle.