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.

Transformation and reduction of equations and systems, normal forms (msc 34C20), Transformation and reduction of equations and systems, normal forms (msc 34K17), Singularities, monodromy, local behavior of solutions, normal forms (msc 34M35), Normal forms (msc 37G05), Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion (msc 37J40), Bifurcation problems (msc 37K50), None of the above, but in MSC2010 section 37Kxx (msc 37K99), Normal forms, center manifold theory, bifurcation theory (msc 37L10), Normal forms (msc 58K50), Normal forms (msc 70K45), Bifurcation (msc 34C23), Bifurcation theory (msc 34K18), Bifurcation (msc 35B32), Local and nonlocal bifurcation theory (msc 37Gxx), Bifurcation theory (msc 37H20), Bifurcation problems (msc 37J20), Bifurcation problems (msc 37K50), Normal forms, center manifold theory, bifurcation theory (msc 37L10), Computational methods for bifurcation problems (msc 37M20), Bifurcations and instability (msc 70K50)
CWI
Modelling, Analysis and Simulation [MAS]

Kuznetsov, Y.A. (1997). Explicit normal form coefficients for all codim 2 bifurcations of equilibria in ODEs. Modelling, Analysis and Simulation [MAS]. CWI.