  Based on the Taylor expansion, we propose a renormalization method for
asymptotic analysis. The standard renormalization group (RG) method for
asymptotic analysis can be derived out from this new method, and hence the
mathematical essence of the RG method is also recovered. The biggest advantage
of the proposed method is that the secular terms in perturbation series are
automatically eliminated, but in usual perturbation theory, we need more
efforts and tricks to eliminate these terms. At the same time, the mathematical
foundation of the method is simple and the logic of the method is very clear,
therefore, it is very easy in practice. As application, we obtain the uniform
valid asymptotic solutions to some problems including vector field, boundary
layer and boundary value problems of nonlinear wave equations. Moreover, we
discuss the normal form theory and reduction equations of dynamical systems.
Furthermore, by combining the topological deformation and the RG method, a
modified method namely the homotopy renormalization method (for simplicity,
HTR) wasproposed to overcome the weaknesses of the standard RG method. In this
HTR method, since there is a freedom to choose the first order approximate
solution in perturbation expansion, we can improve the global solution. In
particular, for those equations including no a small parameter, the HTR method
can also be applied. Some concrete applications including multi-solutions
problems, the forced Duffing equation and the Blasius equation are given.
