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On periodic solutions inside isolating chains

机译：关于隔离链内部的周期解

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We develop a geometric approach to problems concerning the existence of T-periodic solutions of a non-autonomous time-T periodic ordinary differential equation. We consider isolating segments, subsets of the extended phase space of the equation, which in some ways resemble isolating blocks from the theory of isolated invariant sets. The union of several contiguous isolating segments is called an isolating chain. Isolating segments determine some homomorphisms in reduced singular homologies. The main theorem asserts that the Lefschetz number of the composition of the homomorphisms determined by segments such that their union is a periodic isolating chain is equal to the fixed point index of the Poincare map of the equation in the set of initial values of T-periodic solutions contained inside the chain. We give some applications of the theorem to planar polynomial equations. In particular, we prove that the equation (z) over dot = (z) over bar(S) + sin(2)(phi t) = has four nonzero (pi/phi)-periodic solutions provided 0 < phi less than or equal to pi/336. (C) 2000 Academic Press. [References: 17]
机译：我们开发了一种解决非自治时间T周期常微分方程T周期解存在性问题的几何方法。我们考虑隔离段，即方程扩展相空间的子集，在某些方面类似于隔离不变集理论中的隔离块。几个连续的隔离链段的并集称为隔离链。隔离段确定了减少的奇异同调的一些同态。主定理断言，由各段确定的同构的Lefschetz数，使得它们的并集是一个周期性的隔离链，等于该T周期初始值集中方程的庞加莱图的不动点索引解决方案包含在链内。我们将定理应用于平面多项式方程。特别是，我们证明点（s）上bar（S）+ sin（2）（phi t）=上的方程（z）具有四个非零（pi / phi）周期解，且0 hi小于或。等于pi / 336。 （C）2000学术出版社。 [参考：17]

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