Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE's

Explicit computational formulas for the coefficients of the normal forms fo r all codim 2 equilibrium bifurcations of equilibria in autonomous ODEs are derived. They include second-order coefficients for the Bogdanov-Takens bi furcation, third-order coefficients for the cusp and fold-Hopf bifurcations , and coefficients of the fifth-order terms for the generalized Hopf (Bauti n) and double Hopf bifurcations. The formulas are independent on the dimens ion of the phase space and involve only critical eigenvectors of the Jacobi an 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 critica l equilibrium. The normal form coefficients for the fold-Hopf bifurcation i n the "new" Lorenz model are computed using the derived formulas, proving t he existence of a nontrivial invariant set in the system.