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A generalized Lefschetz fixed point theorem and symbolic dynamics in delay equations

Published online by Cambridge University Press:  06 August 2002

BERNHARD LANI-WAYDA  and
ROMAN SRZEDNICKI
BERNHARD LANI-WAYDA
Affiliation:
Mathematisches Institut der Universität Giessen, Arndtstrasse 2, 35392 Giessen, Germany (e-mail: [email protected])
ROMAN SRZEDNICKI
Affiliation:
Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland (e-mail: [email protected])
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Abstract

We prove a generalized version of the Lefschetz fixed point theorem, and use it to obtain a variety of periodic and aperiodic solutions for differential delay equations; in particular, of the type \dot x(t) = f(x(t-1)). Here f:\mathbb R \to \mathbb R is odd and two-periodic, and we obtain both strictly periodic solutions and solutions periodic modulo a multiple of two. The qualitative behavior of solutions can be coded by symbol sequences containing the sequence of levels about which these solutions oscillate.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 4 , August 2002 , pp. 1215 - 1232
Copyright
© 2002 Cambridge University Press

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