: An asymptotic unfolding of a dynamical system near a rest point is a system with additional parameters, such that every one parameter deformation of the original system can be embedded in the unfolding preserving all properties that can be detected by asymptotic methods. Asymptotic unfolding are computed using normal (and hyper normal) form methods. We present the simplified and improved method of computing such unfolding (introduced by Murdock) that can be used in any normal form style. In this thesis we also present the method of multiple Lie bracket method introduced by Kokubu, Oka, and Wang. Then, we demonstrate how they improved their method and then used to solve an unsolved problem in a paper of Baider and Sanders for the unique normal form of BogdanovTakens singularities, where the problem under a very general condition is solved. The idea of quasi-homogeneous normal form theory using new grading functions is introduced, the definition of N-th order normal form is given and some sufficient conditions for the uniqueness of normal forms are derived. Another special case of the unsolved problem in a paper of Baider and Sanders for the unique normal form of BogdanovTakens singularities is also solved. At the end we present our research result on the simplest orbital normal form of generalized Saddle-node case of Bogdanov – Takens singularity. We follow the paper of Baider and Sanders in using the sl(2)- representation of vector fields for obtaining the