  This paper provides for the first time correct third-order homoclinic
predictors in $n$-dimensional ODEs near a generic Bogdanov-Takens bifurcation
point. To achieve this, higher-order time approximations to the nonlinear time
transformation in the Lindstedt-Poincar\'e method are essential. Moreover, a
correct transform between approximations to solutions in the normal form and
approximations to solutions on the parameter-dependent center manifold is
needed. A detailed comparison is done between applying different normal forms
(smooth and orbital), different phase conditions, and different perturbation
methods (regular and Lindstedt-Poincar\'e) to approximate the homoclinic
solution near Bogdanov-Takens points. Examples demonstrating the correctness of
the predictors are given. The new homoclinic predictors are implemented in the
open-source MATLAB/GNU Octave continuation package MatCont.
