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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The standard Sharkovsky cycle coexistence theorem applies to impulsive differential equations: Some notes and beyond
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by Jan Andres PDF
Proc. Amer. Math. Soc. 147 (2019), 1497-1509 Request permission

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.
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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
  • 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
  • © Copyright 2019 American Mathematical Society
  • 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
  • MathSciNet review: 3910415