Lie equations for asymptotic solutions of perturbation problems of ordinary differential equations

Lie theory is applied to perturbation problems of ordinary differential equations to construct approximate solutions and invariant manifolds according to the renormalization group approach of Iwasa and Nozaki ["A method to construct asymptotic solutions invariant under the renormalization group," Prog. Theor. Phys. 116, 605 (2006)]. It is proved that asymptotic behavior of solutions is obtained from the Lie equations even if original equations have no symmetries. Normal forms of the Lie equations are introduced to investigate the existence of invariant manifolds.