Renormalization Reductions for Systems with Delay

TL;DR

This paper extends the renormalization perturbation method to analyze weakly nonlinear systems with delay, deriving reduced models with or without delay depending on the system's delay size, and validates these models analytically and numerically.

Contribution

The paper introduces an extended renormalization method for delay systems, enabling reduction to simpler models with or without delay, depending on the delay magnitude.

Findings

Reduced systems without delay for order-one delay systems.

Reduced systems with delay for large-delay systems.

Analytical and numerical validation of the reduced models.

Abstract

The renormalization method which is a type of perturbation method is extended to a tool to study weakly nonlinear time-delay systems. For systems with order-one delay, we show that the renormalization method leads to reduced systems without delay. For systems with order-one and large-delay, we propose an extended renormalization method which leads to reduced systems with delay. In some examples, the validities of our perturbative results are confirmed analytically and numerically. We also compare our reduced equations with reduced ones obtained by another perturbation method.