Normal form analysis for bifurcations with Huygens symmetry
This thesis presents a study of the dynamics of two coupled oscillators from the point view of equivariant bifurcation theory. It is assumed that the vector field possesses Huygens symmetry and the system undergoes double Hopf bifurcation near 1:1 resonance. Huygens symmetry means that the system is unchanged if the two oscillators are interchanged and the individual oscillators are both unchanged under reflection. For such a system, it is shown that there exist two flow invariant fixed-point subspaces which correspond to the in-phase and anti-phase synchronization of Huygens' clocks. The equivariant normal form is derived using two different methods: the Elphick method and the Hilbert-Weyl Theorem. In the analysis of the normal form, existence of normal mode and mixed mode phased-locked periodic solutions is proven. Using methods from topological degree theory conditions are given for the existence of quasi-periodic solutions on a 3-torus of the 4-dimensional normal form equations for a case in which the generic double Hopf bifurcation has quasi-periodic solutions on a 2-torus. A model example of coupled van der Pol oscillators is explored. In addition, the normal form for the double zero (Bogdanov-Takens) bifurcation with Huygens symmetry is presented.