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A general reduction method for periodic solutions in conservative and reversible systems

  • Jürgen Knobloch
  • André Vanderbauwhede
Article

Abstract

We introduce a general reduction method for the study of periodic solutions near equilibria in autonomous systems which are either conservative or reversible. We impose no restrictions on the linearization at the equilibrium, allowing higher multiplicities and all kinds of resonances. It is shown that the problem reduces to a similar problem for a reduced system, which is itself conservative or reversible, but also has an additionalS1-symmetry. This symmetry allows to immediately write down the bifurcation equations. Moreover, the reduced system can be calculated up to any order by a normal form reduction on the original system. The method of proof combines normal forms with the Liapunov-Schmidt method. A similar approach was already introduced for Hamiltonian systems in [9], and for equivariant systems in [3]; this paper extends the results of these papers to the cases of conservative and reversible systems.

Key words

Conservative and reversible systems periodic orbits normal form theory Liapunov-Schmidt reduction 

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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Jürgen Knobloch
    • 1
  • André Vanderbauwhede
    • 2
  1. 1.Department of MathematicsTechnische Universität IlmenauIlmenauGermany
  2. 2.Department of Pure Mathematics and Computer AlgebraUniversity of GentGentBelgium

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