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Extended Hopf Bifurcation for Abstract Integral Equations at Resonant Eigenvalue

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Abstract

In this paper, we apply an extension of the abstract Hopf bifurcation theorem stated in Jaćimović (Non Anal TMA 73(8):2426–2432, 2010) to abstract integral equations (AIE) and retarded functional differential equations (RFDE). This yields sufficient conditions of what we refer to as extended Hopf bifurcation for AIE and RFDE, in which we have a relaxation of the non-resonance condition on the eigenvalues of the generator of corresponding semigroup. We illustrate our results with an explicit example of a system of two delay differential equations (DDE), undergoing extended Hopf bifurcation at the resonant eigenvalue.

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Acknowledgments

The author acknowledges partial support of the Ministry of Science of Montenegro, grant number 05-1/3-2008. The author would like to thank the anonymous referee for the numerous valuable comments and suggestions.

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put, bb., 81000, Podgorica, Montenegro

    Vladimir Jaćimović

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  1. Vladimir Jaćimović
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Correspondence to Vladimir Jaćimović.

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Jaćimović, V. Extended Hopf Bifurcation for Abstract Integral Equations at Resonant Eigenvalue. J Dyn Control Syst 20, 431–442 (2014). https://doi.org/10.1007/s10883-014-9235-6

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  • DOI: https://doi.org/10.1007/s10883-014-9235-6

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