Explicit Computations of Hopf and Bautin Bifurcations in 2 -Symmetric Systems

By using a homogical method, we drive out computational formulae for normal forms of the Hopf and Bautin bifurcations in  2 -symmetric systems. For practical bifurcation analysis of Hopf and Bautin in a  2 -symmetric system, we can use these formulae to compute the first and the second Lyapunov coefficients, and check whether the bifurcation is degenerate. Furthermore, we can use the formulae of unfolding parameters to decide the topological structures when parameters perturb in a neighborhood of the critical values. So, we construct the relation between the parameters and the structures for Hopf and Bautin bifurcations in any  2 -symmetric systems.