The standard Sharkovsky cycle coexistence theorem applies to impulsive differential equations: Some notes and beyond

Abstract:

We will show that, unlike usual (i.e., nonimpulsive) differential equations, the standard Sharkovsky cycle coexistence theorem applies easily to impulsive, scalar, ordinary differential equations. In fact, there is a one-to-one correspondence between the subharmonic solutions of given orders and periodic points of the same orders of the associated Poincaré translation operators, provided a uniqueness condition is satisfied. Despite the fact that the usage of the Poincaré operators in the context of impulsive differential equations is neither new, nor original, and that the application of the Sharkovsky celebrated theorem becomes in this way rather trivial, as far as we know, an appropriate theorem has not yet been formulated. As a by-product, the relationship of impulsive differential equations to deterministic chaos will also be clarified. In order to demonstrate the merit of the basic idea, some less trivial extensions for discontinuous and multivalued impulses will still be briefly done, along with indicating the situation in the lack of uniqueness.