Transient behavior of the solutions to the second order difference equations by the renormalization method based on Newton-Maclaurin expansion

TL;DR

This paper applies a renormalization method based on Newton-Maclaurin expansion to analyze the transient behavior of second order difference equations, providing a natural alternative to multi-scale methods.

Contribution

It introduces a novel renormalization approach using Newton-Maclaurin expansion to study transient solutions of difference equations, emphasizing its naturalness over existing multi-scale methods.

Findings

Effective description of transient behavior using elementary functions

Demonstrates the method's advantages over multi-scale approaches

Applied to important second order nonlinear difference equations

Abstract

The renormalization method based on the Newton-Maclaurin expansion is applied to study the transient behavior of the solutions to the difference equations as they tend to the steady-states. The key and also natural step is to make the renormalization equations to be continuous such that the elementary functions can be used to describe the transient behavior of the solutions to difference equations. As the concrete examples, we deal with the important second order nonlinear difference equations with a small parameter. The result shows that the method is more natural than the multi-scale method.