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Hopf Bifurcation for an Infinite Delay Functional Equation

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Dynamics of Infinite Dimensional Systems

Part of the book series: NATO ASI Series ((NATO ASI F,volume 37))

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Abstract

In this lecture we discusse a Hopf bifurcation problem for the functional equation

$$ x(t) = F(\alpha ,x_t ),t \in R. $$
(1.1)

Here x is a vector in R n, the parameter α is an element of a finite dimensional real Banach space A, and F is a mapping from Α × BUC(R;R n) into R n. Moreover, F(α, 0) = 0, so x ≡ 0 is a solution of (1.1). In addition we suppose that the linearization of (1.1) has a one-parameter family of nontrivial periodic solutions at a critical value α 0 of the parameter. Our aim is to show that also the nonlinear equation has a oneparameter family of nontrivial periodic solutions for some values of α close to α 0.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Helsinki University of Technology, SF-02150, Espoo 15, Finland

    Olof J. Staffans

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  1. Olof J. Staffans
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Michigan State University, 48824-1027, East Lansing, MI, USA

    Shui-Nee Chow

  2. Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA

    Jack K. Hale

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© 1987 Springer-Verlag Berlin Heidelberg

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Staffans, O.J. (1987). Hopf Bifurcation for an Infinite Delay Functional Equation. In: Chow, SN., Hale, J.K. (eds) Dynamics of Infinite Dimensional Systems. NATO ASI Series, vol 37. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-86458-2_28

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  • DOI: https://doi.org/10.1007/978-3-642-86458-2_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-86460-5

  • Online ISBN: 978-3-642-86458-2

  • eBook Packages: Springer Book Archive

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