Sharkovskii’s theorem, differential inclusions, and beyond. (English) Zbl 1189.34028
Authors’ abstract: We explain why the Poincaré translation operators along the trajectories of upper-Carathéodory differential inclusions do not satisfy the exceptional cases, described in our earlier counter-examples, for upper semicontinuous maps. Such a discussion was stimulated by a recent paper of F. Obersnel and P. Omari, where they show that, for Carathéodory scalar differential equations, the existence of just one subharmonic solution (e.g. of order 2) implies the existence of subharmonics of all orders. We reprove this result alternatively just via a multivalued Poincaré translation operator approach. We also establish its randomized version on the basis of a universal randomization scheme developed recently by the first author.
Reviewer: Francesca Papalini (Ancona)
MSC:
34A60 | Ordinary differential inclusions |
37E15 | Combinatorial dynamics (types of periodic orbits) |
37H10 | Generation, random and stochastic difference and differential equations |
47H04 | Set-valued operators |
47H40 | Random nonlinear operators |
34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |