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Equivariant normal forms for parameterized delay differential equations with applications to bifurcation theory

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Abstract

In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.

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Authors and Affiliations

  1. College of Mathematics and Econometrics, Hu’nan University, Changsha, 410082, P. R. China

    Shang Jiang Guo

  2. Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

    Yu Ming Chen

  3. Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

    Jian Hong Wu

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  1. Shang Jiang Guo
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  2. Yu Ming Chen
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  3. Jian Hong Wu
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Correspondence to Shang Jiang Guo.

Additional information

The first author is supported by NSFC (Grant No. 10971057), the Key Project of Chinese Ministry of Education (Grant No. [2009]41), and by Hu’nan Provincial Natural Science Foundation (Grant No. 10JJ1001), and by the Fundamental Research Funds for the Central Universities, Hu’nan University; the second author is supported by NSERC of Canada and by ERA Program of Ontario; the third author is supported in part by MITACS, CRC, and NSERC of Canada

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Guo, S.J., Chen, Y.M. & Wu, J.H. Equivariant normal forms for parameterized delay differential equations with applications to bifurcation theory. Acta. Math. Sin.-English Ser. 28, 825–856 (2012). https://doi.org/10.1007/s10114-011-9718-2

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  • DOI: https://doi.org/10.1007/s10114-011-9718-2

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