Explicit normal form coefficients for all codim 2 bifurcations of equilibria in ODEs

摘要: In this paper, explicit formulas for the coefficients of normal forms all codim 2 equilibrium bifurcations equilibria in autonomous ODEs are derived. They include second-order Bogdanov-Takens bifurcation, third-order cusp and fold-Hopf bifurcations, fifth-order terms generalized Hopf (Bautin) double bifurcations. The independent on dimension phase space involve only critical eigenvectors Jacobian matrix right-hand sides its transpose, as well multilinear functions from Taylor expansion at equilibrium.