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

TL;DR

This paper extends the renormalization method from differential to difference equations using the Newton-Maclaurin expansion, providing theoretical foundations and applications to quantum oscillators and discrete systems.

Contribution

It introduces a generalized renormalization technique for difference equations, including a homotopy approach for equations without small parameters.

Findings

Derived asymptotic solutions for quantum anharmonic oscillators

Established reductions and invariant manifolds for discrete systems

Proposed a homotopy renormalization method for complex difference equations

Abstract

The renormalization method based on the Taylor expansion for asymptotic analysis of differential equations is generalized to difference equations. The proposed renormalization method is based on the Newton-Maclaurin expansion. Several basic theorems on the renormalization method are proven. Some interesting applications are given, including asymptotic solutions of quantum anharmonic oscillator and discrete boundary layer, the reductions and invariant manifolds of some discrete dynamics systems. Furthermore, the homotopy renormalization method based on the Newton-Maclaurin expansion is proposed and applied to those difference equations including no a small parameter.