  In solving real world systems for higher-codimension bifurcation problems,
one often faces the difficulty in computing the normal form or the focus values
associated with generalized Hopf bifurcation, and the normal form with
unfolding for higher-codimension Bogdanov-Takens bifurcation. The difficulty is
not only coming from the teadious symbolic computation of focus values, but
also due to the restriction on the system parameters, which frequently leads to
failure of the conventional approach used in the computation even for simple
$2$-dimensional nonlinear dynamical systems. In this paper, we use a simple
2-dimensional epidemic model, for which the conventional approach fails in
analyzing the stability of limit cycles arising from Hopf bifurcation, to
illustrate how our method can be efficiently applied to determine the
codimension of Hopf bifurcation. Further, we apply the simplest normal form
theory to consider codimension-3 Bogdanov-Takens bifurcation and present an
efficient one-step transformation approach, compared with the classical
six-step transformation approach to demonstrate the advantage of our method.
