On removing orders from amplitude equations

TL;DR

This work presents a modified renormalization-group method that introduces homogeneous functions at each perturbation order to suppress higher-order terms in amplitude equations, while preserving overall accuracy. A key insight is the existence of a nonlinear core in the amplitude equation that cannot be removed without inducing linear growth, often associated with logarithmic terms in the homogeneous components. The authors demonstrate the approach on scalar ODEs (Duffing and Van der Pol) and on systems (Lotka-Volterra, a two-amplitude system, and the Selkov Hopf model), showing that the amplitude equations can be markedly simplified while maintaining or even improving long-time accuracy. The method relies on tuning a small set of free parameters within the homogeneous functions to minimize error, sometimes achieving performance on par with classical RG. These results suggest broad applicability to nonlinear dynamics and potential extensions to nonlinear PDEs, where simplifying amplitude dynamics can offer substantial computational and analytical benefits.

Abstract

In this paper, we introduce a modified version of the renormalization group (RG) method and test its numerical accuracy. It has been tested on numerous scalar ODEs and systems of ODEs. Our method is primarily motivated by the possibility of simplifying amplitude equations. The key feature of our method is the introduction of a new homogeneous function at each order of the perturbation hierarchy, which is then used to remove terms from the amplitude equations. We have shown that there is a limit to how many terms can be removed, as doing so beyond a certain point would reintroduce linear growth. There is thus a \textit{core} in the amplitude equation, which consists of the terms that cannot be removed while avoiding linear growth. Using our modified RG method, higher accuracy can also be achieved while maintaining the same level of complexity in the amplitude equation.

Figures (12)

• Figure 1: Plots of a high precision numerical solution (\ref{['eq1']}) and the modified RG solution (\ref{['eq51']}), (\ref{['eq52']}) for $t\in[9900,10000]$.
• Figure 2: Plot of the difference between the high precision numerical solution to (\ref{['eq1']}) and the RG solution (\ref{['eq51']}), (\ref{['eq52']}) using the homogeneous functions for $t\in[0,10^4]$.
• Figure 3: Comparing the errors of the modified RG solution (\ref{['eq50']}), (\ref{['eq51']}) fig. \ref{['fig3a']} and the classical RG solution (\ref{['eq53']}), (\ref{['eq54']}) fig. \ref{['fig3b']} against high precision numerical solution. On the left, the timeline spans through $t\in [0,200]$ and on the right, it goes through $t\in [9800,10^4]$.
• Figure 4: Comparing the errors of two RG solutions of the Van der Pol oscillator (\ref{['eq55']}), the classical (\ref{['eq91']}) (left) and modified (\ref{['eq55.2']}) (right) with all parameters set to zero. The error is against the high precision numerical solution. In both cases, the timeline spans through $t\in [9900,10^4]$.
• Figure 5: Comparing the error of the classical RG solution using (\ref{['eq91']}) with the modified RG solution using (\ref{['eq55.3']}), where the free parameter are (\ref{['eq93']}).