The renormalization method from continuous to discrete dynamical systems: asymptotic solutions, reductions and invariant manifolds

摘要

The renormalization method based on the Taylor expansion for asymptoticanalysis of differential equations is generalized to difference equations. Theproposed renormalization method is based on the Newton-Maclaurin expansion.Several basic theorems on the renormalization method are proven. Someinteresting applications are given, including asymptotic solutions of quantumanharmonic oscillator and discrete boundary layer, the reductions and invariantmanifolds of some discrete dynamics systems. Furthermore, the homotopyrenormalization method based on the Newton-Maclaurin expansion is proposed andapplied to those difference equations including no a small parameter.