Explicit transversality conditions and local bifurcation diagrams for Bogdanov–Takens bifurcation on center manifolds

Highlights: • Parametrization for homological computation of the Bogdanov–Takens normal form is reconsidered. • Simple formulas are obtained for parameter transformation for enduring equilibria. • The computation can be limited in certain subspaces for generalized $Z_2$-symmetric systems. • The results enable quick check of transversality and depiction of bifurcation diagrams. -- Abstract: This application-oriented study is concerned with the derivation of parameter-dependent normal forms for the codimension-two Bogdanov–Takens bifurcation of $n$-dimensional, $m$-parameterized systems on the basis of the homological method. In the case of an enduring equilibrium, simple formulas are obtained for the transformation of parameters, enabling the formulation of explicit transversality conditions and bifurcation diagrams to at most the second order. Moreover, in $Z_2$-symmetric systems, the calculation can be further limited within certain subspaces. In the general case, existing results are re-derived, and a revision necessary for determining the bifurcation diagrams to the second order is then provided. These results facilitate the derivation of normal forms, check of transversality and depiction of bifurcation diagrams for the Bogdanov–Takens bifurcation.