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Symmetry Breaking in Dynamical Systems

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Nonlinear Dynamical Systems and Chaos

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 19))

Abstract

Symmetry breaking bifurcations and dynamical systems have obtained a lot of attention over the last years. This has several reasons: real world applications give rise to systems with symmetry, steady state solutions and periodic orbits may have interesting patterns, symmetry changes the notion of structural stability and introduces degeneracies into the systems as well as geometric simplifications. Therefore symmetric systems are attractive to those who study specific applications as well as to those who are interested in a the abstract theory of dynamical systems. Dynamical systems fall into two classes, those with continuous time and those with discrete time. In this paper we study only the continuous case, although the discrete case is as interesting as the continuous one. Many global results were obtained for the discrete case. Our emphasis are heteroclinic cycles and some mechanisms to create them. We do not pursue the question of stability. Of course many studies have been made to give conditions which imply the existence and stability of such cycles. In contrast to systems without symmetry heteroclinic cycles can be structurally stable in the symmetric case. Sometimes the various solutions on the cycle get mapped onto each other by group elements. Then this cycle will reduce to a homoclinic orbit if we project the equation onto the orbit space. Therefore techniques to study homoclinic bifurcations become available. In recent years some efforts have been made to understand the behaviour of dynamical systems near points where the symmetry of the system was perturbed by outside influences. This can lead to very complicated dynamical behaviour, as was pointed out by several authors. We will discuss some of the technical difficulties which arise in these problems. Then we will review some recent results on a geometric approach to this problem near steady state bifurcation points.

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Authors and Affiliations

  1. WIAS Berlin, Mohrenstr. 39, D-10117, Berlin, Germany

    Reiner Lauterbach

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  1. Reiner Lauterbach
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Editor information

Editors and Affiliations

  1. Institute of Mathematics and Computer Science, University of Groningen, P.O. Box 800, 9700 AV, Groningen, The Netherlands

    H. W. Broer , I. Hoveijn  & F. Takens ,  & 

  2. Faculty of Mathematics and Computer Science, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    S. A. van Gils

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© 1996 Springer Basel AG

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Lauterbach, R. (1996). Symmetry Breaking in Dynamical Systems. In: Broer, H.W., van Gils, S.A., Hoveijn, I., Takens, F. (eds) Nonlinear Dynamical Systems and Chaos. Progress in Nonlinear Differential Equations and Their Applications, vol 19. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7518-9_6

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  • DOI: https://doi.org/10.1007/978-3-0348-7518-9_6

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-7520-2

  • Online ISBN: 978-3-0348-7518-9

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