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Homoclinic Flip Bifurcations in Conservative Reversible Systems

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Recent Trends in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 35))

Abstract

In this paper, flip bifurcations of homoclinic orbits in conservative reversible systems are analyzed. In such systems, orbit-flip and inclination-flip bifurcations occur simultaneously. It is shown that multi-pulses either do not bifurcate at all at flip bifurcation points or else bifurcate simultaneously to both sides of the bifurcation point. An application to a fifth-order model of water waves is given to illustrate the results, and open problems regarding the PDE stability of multi-pulses are outlined.

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Notes

  1. 1.

    The Hessian of the energy restricted to the eigenspace associated with a quadruplet off the imaginary axis must decrease and increase in two transverse planes; thus eigenvalues can leave the imaginary axis only when the energy restricted to their combined eigenspace is indefinite, that is, the eigenvalues have opposite Krein signatures; see [7] and references therein.

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Acknowledgements

This paper is dedicated to Jürgen Scheurle on the occasion of his 60th birthday: I am deeply grateful for his expressions of encouragement and support when I began my career as a graduate student and postdoc.

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Authors and Affiliations

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

    Björn Sandstede

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  1. Björn Sandstede
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Correspondence to Björn Sandstede .

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Editors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Andreas Johann

  2. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Hans-Peter Kruse

  3. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Florian Rupp

  4. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Stephan Schmitz

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Sandstede, B. (2013). Homoclinic Flip Bifurcations in Conservative Reversible Systems. In: Johann, A., Kruse, HP., Rupp, F., Schmitz, S. (eds) Recent Trends in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 35. Springer, Basel. https://doi.org/10.1007/978-3-0348-0451-6_6

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