Normal form computations in CONTENT

CONTENT is an interactive environment for detection, analysis, and continuation of various bifurcations of ODEs, iterated maps, and other dynamical systems. An important part of the analysis of local bifurcations is the computation of their normal forms at the critical parameter values. This means restricting the system to the corresponding center manifold and then normalizing the system on the center manifold by appropriate changes of variables, parameters, and time. The algorithms used in CONTENT to compute numerically the coefficients of the normal forms for

fold,
Hopf,
cusp, and
Bogdanov-Takens


bifurcations of ODEs, as well as for

fold,
flip, and
Neimark-Sacker


bifurcations of iterated maps, will be presented in detail. An important feature of these algorithms is that no preliminary transformation of the linear part to the Jordan normal form is applied.