On Periodic Solutions inside Isolating Chains

Abstract

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 Poincaré 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 ż=z5+sin2(φt)z has four nonzero (π/φ)-periodic solutions provided 0<φ⩽π/336.