Periodic segment implies infinitely many periodic solutions

Marzantowicz, Wacław; Wójcik, Klaudiusz

doi:10.1090/S0002-9939-07-08750-3

Periodic segment implies infinitely many periodic solutions
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by Wacław Marzantowicz and Klaudiusz Wójcik

Proc. Amer. Math. Soc. 135 (2007), 2637-2647

DOI: https://doi.org/10.1090/S0002-9939-07-08750-3

Published electronically: March 21, 2007

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Abstract:

In this note we show that the existence of a periodic segment for a non-autonomous ODE with periodic coefficients implies the existence of infinitely many periodic solutions inside this segment provided that a sequence of Lefschetz numbers of iterations of an associated map is not constant. In the case when this sequence is bounded we have to impose a geometric condition on the segment to get solutions by use of symbolic dynamics.

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