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This paper initiates the classification, up to symmetry-covariant contact equivalence, of perturbations of local Hopf bifurcation problems which do not satisfy the classical non-degeneracy conditions. The only remaining hypothesis is that +i should be simple eigenvalues of the linearized right-hand side at criticality. Then the Lyapunov-Schmidt method… (More)

This paper investigates generic bifurcations from a D m-invariant equilibrium of a D m-symmetric dynamical system, for m = 3 or 4, near points of codimension-2 steady-state mode interactions. The center manifold is isomorphic to IR 3 or IR 4 and is non-irreducible. Depending on the group representation, in the unfolding of the linearization we nd:… (More)

- W. F. Langford
- 1998

A codimension-three bifurcation, characterized by a pair of purely imaginary eigenvalues and a nonsemisimple double zero eigenvalue, arises in the study of a pair of weakly coupled nonlinear oscillators with Z 2 Z 2 symmetry. The methodology is based on Arnold's ideas of versal deformations of matrices for the linear analysis, and Poincar e normal forms for… (More)

Double Hopf bifurcations have been studied prior to this work in the generic nonresonant case and in certain strongly resonant cases, including 1:1 resonance. In this paper, the case of symmetrically coupled identical oscillators, motivated by the classic problem of synchronization of Huygens' clocks, is studied using the codimension-three Elphick–Huygens… (More)

A mathematical model of convection of a Boussinesq fluid in a rotating spherical shell is analyzed using numerical computations guided by bifurcation theory. The fluid is differentially heated on its inner spherical surface, with the temperature increasing from both poles to a maximum at the equator. The model is assumed to be both rotationally symmetric… (More)

This paper presents a study of the effects of symmetry on the generic bifurca-tion at a double-zero eigenvalue that was first investigated by Bogdanov and Takens. Two different symmetry groups are considered: Huygens symmetry and odd-Huygens symmetry. Here Huygens symmetry means that the system is equivariant under permutation of the two state variables.… (More)

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