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A Multivalued Version of the Block–Sharkovsky Theorem Applicable to Differential Equations on the Circle

Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. Listopadu 12, 771 46 Olomouc, Czech Republic

E-mail Address: jan.andres@upol.cz

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Karel Pastor

Department of Mathematics, Faculty of Education, Palacký University, Žižkovo Náměstí 5, 771 40 Olomouc, Czech Republic

E-mail Address: karel.pastor@upol.cz

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https://doi.org/10.1142/S0218127418501420Cited by:0 (Source: Crossref)

Abstract

A multivalued version of the well-known (Sharkovsky type) Block cycle coexistence theorem is, on the basis of our former results, completed and applied to differential equations and inclusions. The deterministic results are also randomized which allows us, besides other things, to eliminate some exceptional absent periodic dynamics. In this way, instead of at most two possible deterministic exceptional cases (w.r.t. the standard Block theorem), only one possible random exception can occur, provided the forcing period $n = 2^{m} \cdot 3$ $n = 2^{m} \cdot 3$ , $m \in ℕ \cup {0}$ . On the other hand, the application to random differential equations and inclusions is not so effective in general.

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