Many phenomena in chemistry , physics and engineering can be modeled by parametric nonlinear differential systems . These systems demonstrate complicated dynamics , when the parameters reach certain singular values . Therefore , it is important to understand their dynamics near the critical values . Normal form theory is one of the most efficient methods for the local bifurcation analysis of such systems . The main idea is to use a nonlinear change of coordinates to convert a given vector field to its simplest form , called the simplest normal form . The resulting system shares certain qualitative properties of the original system . Here , we mean by {\\it qualitative properties} as those properties invariant under our permissible changes of coordinates . In this thesis , we discuss normal forms for those systems whose linear part has a pair of imaginary eigenvalues (called Hopf singularity) as well as those with a zero eigenvalue along with a pair of imaginary eigenvalues (so called Hopf-zero singularity) . Derivation of the focus values are also considered as an application of our computer program implementation in Maple of the parametric Hopf singularity with symbolic coefficients . Given our approach , the first (well-known) and second order focus values are readily derived and presented . Recently , the simplest normal form for Hopf-zero singularity has been obtained through a representation of \\(sl(2)\\) Lie algebra over the space of all dir=rtl align=center In the existing literature , many mathematical models have been introduced in order for the dynamics study of the HIV-1 virus . In this thesis , we analyze an ordinary differential equation system that models the fighting of the HIV-1 virus with a genetically modified virus . This is to continue a previous result on an HIV-1 therapy model , by fighting the HIV-1 virus by injecting another virus into the infected patient . Here , a modification of the model is proposed , that is to add a constant \\(\\eta\\) into the recombinant virus equation . The associated dynamics is studied in details . It is showed that an increase in the constant \\(\\eta\\) greatly increases the Hopf critical value . A numerical example is provided to demonstrate the bifurcation direction and stability . Here , a normal form computational approach is applied and an accurate estimates for the amplitudes and the periods of the bifurcated limit cycles is given . Numerical simulations are performed in order to confirm the theoretical results . Finally , it is concluded that any increase in \\(\\eta\\) is benefic