Hopf Bifurcations in Problems with O(2) Symmetry: Canonical Coordinates Transformation

Hopf bifurcations in problems with O(2) symmetry are considered. In these problems, the Jacobian matrix is always singular at the circle of Z2 symmetric steady state solutions. While a couple of imaginary eigenvalue cross the imaginary axis, the Hopf bifurcation is not of standard type. The canonical coordinates transformation is used for removing the zero eigenvalue and converting the problem into the standard form. The method is applied to a system of ordinary differential equations on C with many parameters and the stable solutions are obtained using the centre manifold reduction. Further symmetry breaking bifurcation is obtained on periodic solutions, leading to modulated travelling waves solutions.