On a certain renormalization group method

In this paper, a mathematical study of the renormalization group method, recently introduced by Chen, Goldenfeld, and Oono [Phys. Rev. E 54, 376 (1997); Phys. Rev. Lett. 73, 1311 (1994), NATO Adv. Study Inst. Ser., Ser. B 284, 375 (1991)] is given for the case of autonomous nonlinear systems of differential equations. We also observe that the approximation results obtained by this method by Chen, Goldenfeld, and Ono [NATO Adv. Study Inst. Ser., Ser. B 284, 375 (1991)] are valid over long time intervals. Moreover, a connection between this method and the classical Poincare-Dulac normal forms and the averaging method is briefly discussed. (C) 2000 American Institute of Physics. [S0022-2488(00)02705-5].