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Reversible Equivariant Hopf Bifurcation

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Abstract.

In this paper we study codimension-one Hopf bifurcation from symmetric equilibrium points in reversible equivariant vector fields. Such bifurcations are characterized by a doubly degenerate pair of purely imaginary eigenvalues of the linearization of the vector field at the equilibrium point. The eigenvalue movements near such a degeneracy typically follow one of three scenarios: splitting (from two pairs of imaginary eigenvalues to a quadruplet on the complex plane), passing (on the imaginary axis), or crossing (a quadruplet crossing the imaginary axis). We give a complete description of the behaviour of reversible periodic orbits in the vicinity of such a bifurcation point. For non-reversible periodic solutions, in the case of Hopf bifurcation with crossing eigenvalues, we obtain a generalization of the equivariant Hopf Theorem.

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References

  1. Arnold, V.I.: Geometrical methods in the theory of ordinary differential equations, 2nd edn. Grundlehren der Mathematischen Wissenschaften 250, Springer-Verlag, New York, 1988

  2. Bridges, T.J., Furter, J.E.: Singularity theory and equivariant symplectic maps. Lec. Notes Math. 1558, Springer-Verlag, Berlin, 1993

  3. Buono, P.-L., Lamb, J.S.W., Roberts, M.: Bifurcation and branching of equilibria in reversible equivariant vector fields. In preparation

  4. Bredon, G.E.: Introduction to compact transformation groups. Pure Appl. Math. 46, Academic Press, New York, 1972

  5. Dellnitz, M., Melbourne, I., Marsden, J.E.: Generic bifurcation of Hamiltonian vector fields with symmetry. Nonlinearity 5, 979–996 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dellnitz, M., Scheurle, J.: Eigenvalue movement for a class of reversible Hamiltonian systems with three degrees of freedom. In: Dynamics, bifurcation and symmetry (Cargèse, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 437, Kluwer Acad. Publ. Dordrecht, 1994, pp. 105–110

  7. Fiedler, B.: Global bifurcation of periodic solutions with symmetry. Lec. Notes Math. 1309, Springer-Verlag, Berlin, 1988

  8. Golubitsky, M., Marsden, J.E., Stewart, I., Dellnitz, M.: The constrained Liapunov-Schmidt procedure and periodic orbits. In: Normal forms and homoclinic chaos (Waterloo, ON, 1992). Fields Inst. Commun. 4, Am. Math. Soc. Providence, RI, 1995, pp. 81–127

  9. Golubitsky, M., Stewart, I: Hopf bifurcation in the presence of symmetry. Arch. Rational Mech. Anal. 87, 107–165 (1985)

    MathSciNet  MATH  Google Scholar 

  10. Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and groups in bifurcation theory. Vol. II, Springer-Verlag, New York, 1988

  11. Hermans, J.: A symmetric sphere rolling on a surface. Nonlinearity 8, 493–515 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hoveijn, I.: Versal deformations and normal forms for reversible and Hamiltonian linear systems. J. Differential Equations 126, 408–442 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hoveijn, I., Lamb, J.S.W., Roberts, M.: Normal forms and unfoldings of linear systems in the eigenspaces of (anti-)automorphisms of order two. J. Differential Equations 190, 182–213 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hoveijn, I., Lamb, J.S.W., Roberts, M.: Reversible equivariant linear systems: normal forms and unfoldings. In preparation

  15. Lamb, J.S.W., Roberts, M.: Reversible equivariant linear systems. J. Differential Equations 159, 239–279 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. van der Meer, J.-C.: The Hamiltonian Hopf bifurcation. Lecture Notes in Mathematics 1160, Springer-Verlag, Berlin, 1985

  17. van der Meer, J.-C.: Hamiltonian Hopf bifurcation with symmetry. Nonlinearity 3, 1041–1056 (1990)

    Article  MATH  Google Scholar 

  18. Melbourne, I., Dellnitz, M.: Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group. Math. Proc. Cambridge Philos. Soc. 114, 235–268 (1993)

    MathSciNet  MATH  Google Scholar 

  19. Melbourne, I.: Versal unfoldings of equivariant linear Hamiltonian vector fields. Math. Proc. Cambridge Philos. Soc. 114, 559–573 (1993)

    MathSciNet  MATH  Google Scholar 

  20. Meyer, K.R.: Bibliographic notes on generic bifurcations in Hamiltonian systems. In: Multiparameter bifurcation theory. Contemp. Math. 56, Am. Math. Soc. Providence, RI, 1986, pp. 373–381

  21. Montaldi, J.: Perturbing a symmetric resonance: the magnetic spherical pendulum. In: Symmetry and perturbation theory (Rome, 1998), World Sci. Publishing, River Edge, NJ, 1999, pp. 218–229

  22. Vanderbauwhede, A.: Hopf bifurcation for equivariant conservative and time-reversible systems. Proc. Roy. Soc. Edinburgh Sect. A 116, 103–128 (1990)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. IBILCE-UNESP, São José do Rio Preto, CEP 15054-000, Brazil

    Claudio A. Buzzi

  2. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    Jeroen S.W. Lamb

Authors
  1. Claudio A. Buzzi
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  2. Jeroen S.W. Lamb
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Communicated by P. J. Holmes

Acknowledgement The material in this paper was developed during a sabbatical of CAB at Imperial College London. CAB would like to thank JSWL and his colleagues for their hospitality, and FAPESP-Brazil, Procad/Capes-Brazil, and the EPSRC-UK for financial support under the grants 01/01914-9, 0092/01-0 and GR/R93025/01. The research of JSWL has been supported in part by the Nuffield Foundation (UK), the Royal Society (UK), and the EPSRC-UK (through an Advanced Research Fellowship and GR/R93025/01). We would like to thank IMPA (Rio de Janeiro) and IMECC-UNICAMP (Campinas) for their hospitality during visits in which this work was finalized.

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Buzzi, C., Lamb, J. Reversible Equivariant Hopf Bifurcation. Arch. Rational Mech. Anal. 175, 39–84 (2005). https://doi.org/10.1007/s00205-004-0337-2

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