Bifurcation analysis for spherically symmetric systems using invariant theory

We study a degenerate steady state bifurcation problem with spherical symmetry. This singularity, with the five dimensional irreducible action ofO(3), has been studied by several authors for codimensions up to 2. We look at the case where the topological codimension is 3, theC∞-codimension is 5. We find a tertiary Hopf bifurcation and a heteroclinic orbit. Our analysis does not use any specific properties of the five dimensional representation and can in principle be used for higher representations as well. The computations are based on invariant theory and orbit space reduction.

[1]  M. Golubitsky,et al.  Pattern selection with O(3) symmetry , 1984 .

[2]  James A. Yorke,et al.  Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation , 1982 .

[3]  Gerald W. Schwarz.  SMOOTH FUNCTIONS INVARIANT UNDER THE ACTION OF A COMPACT LIE GROUP , 1975 .

[4]  Martin Golubitsky,et al.  A Theory for Imperfect Bifurcation via Singularity Theory. , 1979 .

[5]  D. Armbruster,et al.  Heteroclinic orbits in a spherically invariant system , 1991 .

[6]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[7]  P. Chossat,et al.  Steady-State bifurcation with 0(3)-Symmetry , 1991 .

[8]  B. Fiedler,et al.  Dynamics of bifurcations for variational problems with o(3) equivariance: A conley index approach , 1992 .

[9]  Structurally stable heteroclinic cycles in a system with O(3) symmetry , 1991 .

[10]  M. Golubitsky,et al.  Bifurcations with O(3) symmetry including applications to the Bénard problem , 1982 .

[11]  Mode Interactions With Symmetry , 1995 .

[12]  Tammo tom Dieck,et al.  Representations of Compact Lie Groups , 1985 .

[13]  Exclusion of Relative Equilibria , 1994 .

[14]  Tim Poston,et al.  Post-buckling behavior of a non-linearly hyperelastic thin rod with cross-section invariant under the dihedral group Dn , 1985 .

[15]  J. Alexander,et al.  GLOBAL BIFURCATIONS OF PERIODIC ORBITS. , 1978 .

[16]  Valentin Poènaru,et al.  Singularités C「上∞上」 en présence de symétrie : en particulier en présence de la symétrie d'un groupe de Lie compact , 1976 .

[17]  Dynamics near steady state bifurcations in problems with spherical symmetry , 1991 .

[18]  G. L. D. Reis,et al.  Structural stability of equivariant vector fields on two-manifolds , 1984 .

[19]  Martin Golubitsky,et al.  Imperfect bifurcation in the presence of symmetry , 1979 .

[20]  Kenneth R. Meyer,et al.  Singularities and Groups in Bifurcation Theory. Volume II (Martin Golubitsky, Ian Stewart, and David G. Schaeffer) , 1989, SIAM Rev..