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A new characterization of periodic oscillations in periodic difference equations. (English) Zbl 1271.39012
Summary: We characterize periodic solutions of \(p\)-periodic difference equations. We classify the periods into multiples of \(p\) and nonmultiples of \(p\). We show that the elements of the set of multiples of \(p\) follow the well-known Sharkovsky’s ordering multiplied by \(p\). On the other hand, we show that the elements of the set \(\Gamma_p\) of nonmultiples of \(p\) are independent in their existence. Moreover, we show the existence of a \(p\)-periodic difference equation with infinite \(\Gamma_p\)-set in which the maps are defined on a compact domain and agree exactly on a countable set. Based on the proposed classification, we give a refinement of A. N. Sharkovsky’s theorem [“Coexistence of cycles of a continuous map of the line into itself”, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 1263–1273 (1995)] for periodic difference equations.

MSC:
39A21 Oscillation theory for difference equations
39A23 Periodic solutions of difference equations
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