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Limiting Profiles for Solutions of Differential-Delay Equations

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Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1822)

Abstract

This expository paper discusses the ”limiting profile” as ε approaches zero of the graph of x ε , where x ε is a periodic solution either of ε x ε (t) = f(x ε (t), x ε (t - 1)) or of ε x ε (t) = f(x ε (t), x ε (t - r)), r = r(x ε (t)), and f and r are given functions. A variety of theorems from the literature are summarized and proofs of some of the simpler results are sketched. The paper is divided into four sections. The first section gives some background material on fixed point theory, degree theory and the fixed point index. The second section discusses the equation ε x′(t) = f(x(t), x(t - 1)) and treats the family of examples f(x, y) = -x -μ y + y 3 , μ > 1. The final two sections give a sampling of theorems concerning ε x′(t) = f(x(t), x(t - r)), r = r(x(t)), and attempt to give some indication of the techniques involved in the analysis.

Keywords

  • Banach Space
  • Periodic Solution
  • Open Subset
  • Point Theorem
  • Open Neighborhood

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Roger D. Nussbaum .

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© 2003 Springer-Verlag Berlin Heidelberg

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Nussbaum, R.D. (2003). Limiting Profiles for Solutions of Differential-Delay Equations. In: Macki, J.W., Zecca, P. (eds) Dynamical Systems. Lecture Notes in Mathematics, vol 1822. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45204-1_5

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  • DOI: https://doi.org/10.1007/978-3-540-45204-1_5

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40786-7

  • Online ISBN: 978-3-540-45204-1

  • eBook Packages: Springer Book Archive