A new renormalization method for the asymptotic solution of multiple scale singular perturbation problems

This thesis considers the asymptotic integration of special classes of initial value problems involving a nonlinear regular perturbation scaled by a small parameter &epsis; > 0. For t = $O$(1/&epsis;), most of these problems were classically solved by using either the method of averaging or of multiple scales to exorcise secular terms that arise in the natural power series procedure. We present higher order asymptotic approximations by the methods of multiple scales and averaging for weakly nonlinear oscillators. A less well-known invariance condition method for weakly nonlinear vector systems with slowly varying coefficients is formulated in terms of matched asymptotic expansions to obtain higher order asymptotic approximations. Our main result is the construction of a new renormalization method for solving multiple scale singular perturbation problems. For weakly nonlinear vector systems, we derive anew renormalization ansatz that is straightforward and effective. Moreover, it indicates what problems might occur in providing the asymptotic solution on very long time intervals.