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Fixed points of the area preserving Poincaré maps on two-manifolds


Author: Klaudiusz Wójcik
Journal: Proc. Amer. Math. Soc. 145 (2017), 5223-5233
MSC (2010): Primary 34C25; Secondary 37B30
DOI: https://doi.org/10.1090/proc/13642
Published electronically: June 16, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the number of fixed points of the area preserving
Poincaré map $ P$ associated to periodic in time ODE's on two-manifolds. We prove the fixed point index formula for the Poincaré map based on the method of periodic isolating segments. As the application we show that the $ 1$-periodic hamiltonian planar system

$\displaystyle \dot {z}=\overline {z}^n+e^{2\pi i t}\overline {z}^l $

has at least $ n+1$ non-zero $ 1$-periodic solutions provided that $ l>n\geq 1$.

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Additional Information

Klaudiusz Wójcik
Affiliation: Department of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
Email: Klaudiusz.Wojcik@uj.edu.pl

DOI: https://doi.org/10.1090/proc/13642
Keywords: Lefschetz number, isolating blocks, periodic segments, fixed point index, planar hamiltonian systems, Poincar\'e map
Received by editor(s): October 9, 2015
Received by editor(s) in revised form: December 29, 2016
Published electronically: June 16, 2017
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society

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