A version of Sharkovskii’s theorem for differential equations

Abstract:

We present a version of the Sharkovskii cycle coexistence theorem for differential equations. Our earlier applicable version is extended here to hold with the exception of at most two orbits. This result, which (because of counter-examples) cannot be improved, is then applied to ordinary differential equations and inclusions. In particular, if a time-periodic differential equation has $n$-periodic solutions with $n \not = 2^m$, for all $m \in {\mathbb N}$, then infinitely many subharmonics coexist.