A Hopf bifurcation with spherical symmetry

  • Hermann Haaf
  • Mark Roberts
  • Ian Stewart
Original Papers

Summary

We provide a theoretical analysis of a Hopf bifurcation that can occur in systems with spherical geometry, based on the general theory of Hopf bifurcation in the presence of symmetry. In this particular bifurcation the imaginary eigenspace is a direct sum of two copies of the 5-dimensional irreducible representation of the groupSO(3). The same bifurcation has been studied by looss and Rossi (1988), using extensive computer-assisted calculations. Here we describe a simpler and more conceptual approach in which the representation ofSO(3) is realised as its conjugation action on the space of symmetric traceless 3 × 3 matrices. We prove the generic existence of five types of symmetry-breaking oscillation: two rotating waves and three standing waves. We analyse the stabilities of the bifurcating branches, describe the restrictions of the dynamics to various fixed-point spaces of subgroups ofSO(3), and discuss possible degeneracies in the stability conditions.

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References

  1. S. Chandrasekhar,Ellipsoidal Figures of Equilibrium, rev. ed., Dover, New York 1987.Google Scholar
  2. M. J. Field,Equivariant dynamical systems, Trans. Amer. Math. Soc.259, 185–205 (1980).Google Scholar
  3. M. J. Field,Local structure for equivariant dynamics, inSingularity Theory and its Applications, Warwick 1989, Part II (eds. R. M. Roberts and I. N. Stewart), Lect. Notes in Math. 1463, Springer-Verlag, Heidelberg 1991, pp. 142–166.Google Scholar
  4. S. van Gils and M. Golubitsky.A torus bifurcation with symmetry, Dynamics and Diff. Eqns.2, 133–162 (1990).Google Scholar
  5. M. Golubitsky and R. M. Roberts,A classification of degenerate Hopf bifurcations with O(2)symmetry, J. Diff. Eqns.69, 216–264 (1987).Google Scholar
  6. M. Golubitsky and I. N. Stewart,Hopf bifurcation in the presence of symmetry, Arch. Rat. Mech. Anal.87, 107–165 (1985).Google Scholar
  7. M. Golubitsky and I. N. Stewart,Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators, inMultiparameter Bifurcation Theory (eds. M. Golubitsky and J. Guckenheimer), Contemporary Math. 56, Amer. Math. Soc., Providence, RI 1986, pp. 131–173.Google Scholar
  8. M. Golubitsky, I. N. Stewart and D. G. Schaeffer,Singularities and Groups in Bifurcation Theory, Vol. II, Appl. Math. Sci. Series 69, Springer-Verlag, New York 1988.Google Scholar
  9. G. Iooss and M. Rossi,Hopf bifurcation in the presence of spherical symmetry: Analytical Results, Preprint, University of Nice 1988.Google Scholar
  10. K. Kirchgässner,Instability in fluid mechanics, Numerical Solution of Partial Differential Equations III (Proc. 3rd Symposium (SYNSPADE), Univ. of Maryland, College Park MD 1975), Academic Press, New York 1976, pp. 349–371.Google Scholar
  11. K. Kirchgässner,Exotische Lösungen des Bénardschen Problems, Math. Meth. Appl. Sci.1, 453–467 (1979).Google Scholar
  12. J. A. Montaldi, R. M. Roberts and I. N. Stewart,Stability of nonlinear normal modes of Hamiltonian systems with symmetry, Nonlinearity3, 731–772 (1990).Google Scholar
  13. C. Procesi,The invariant theory of n × n matrices, Advances in Math.19, 306–381 (1976).Google Scholar
  14. D. H. Sattinger,Branching in the presence of symmetry, CBMS-NSF Conference Notes 40. SIAM Philadelphia 1983.Google Scholar
  15. A. J. M. Spencer,Theory of Invariants, in C. Bringen (ed.),Continuum Physics, Vol. 1, Part III, Academic Press, New York 1971.Google Scholar
  16. A. J. M. Spencer and R. S. Rivlin,The theory of matrix polynomials and its application to the mechanics of isotropic continua, Arch. Rat. Mech. Anal.2, 309–336 (1959).Google Scholar
  17. I. N. Stewart,Stability of periodic solutions in symmetric Hopf bifurcation, Dyn. Stab. Sys.2, 149–165 (1988).Google Scholar

Copyright information

© Birkhäuser Verlag 1992

Authors and Affiliations

  • Hermann Haaf
    • 1
  • Mark Roberts
    • 1
  • Ian Stewart
    • 1
  1. 1.Nonlinear Systems Laboratory, Mathematics InstituteUniversity of WarwickCoventryEngland

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