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

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