Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Jan Andres
Journal: Proc. Amer. Math. Soc. 147 (2019), 1497-1509
MSC (2010): Primary 34B37, 34C28; Secondary 34C25, 37E05
DOI: https://doi.org/10.1090/proc/14387
Published electronically: January 9, 2019
MathSciNet review: 3910415
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 34B37, 34C28, 34C25, 37E05

Retrieve articles in all journals with MSC (2010): 34B37, 34C28, 34C25, 37E05


Additional Information

Jan Andres
Affiliation: Faculty of Science, Department of Mathematical Analysis and Applications of Mathematics, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
MR Author ID: 222871
Email: jan.andres@upol.cz

Keywords: Sharkovsky’s theorem, coexistence of subharmonics, impulsive differential equations, Poincaré translation operators, Devaney’s chaos, (dis)continuous impulses, multivalued impulses, lack of uniqueness
Received by editor(s): May 22, 2018
Published electronically: January 9, 2019
Additional Notes: The author was supported by the grant IGA_PrF_2018_024 “Mathematical Models” of the Internal Grant Agency of Palacký University in Olomouc.
Communicated by: Wenxian Shen
Article copyright: © Copyright 2019 American Mathematical Society