Binomial expansion for a linearized map with memory. A Final comment on: "Criticality and the fractal structure of -5/3 turbulent cascades"

TL;DR

This paper provides a detailed analysis of a linearized model for turbulence energy transfer, deriving a general formula for its evolution and revealing a fractal structure in its characteristic polynomial calculations.

Contribution

It introduces a binomial expansion approach to analyze the linearized turbulence model and demonstrates the fractal nature of the eigenvalue computation process.

Findings

Derived a general formula for the model's evolution at each step.

Established a connection between matrix products and fractal objects.

Confirmed the fractal structure in the characteristic polynomial calculations.

Abstract

In a recent paper by Cabrera et al (Chaos, Solitons and Fractals 2021 146 110876), a linearization of DRM differences equation, (Delayed Regulation Model), has been proposed as a scheme to explain transfer of energy through different scales in turbulence. They claim that this apparently simple model, by replication of Kolmogorov power law of k ^ -5/3 scaling, remarks a key mechanism of behaviour for more complex systems. Their proposal requires computation of several products of random matrices, nevertheless they only offer an onset of time evolution or an asymptotic approximation to all them. Also it is suggested a fractal nature in the process of calculating the successive characteristic polynomials of these products or the eigenvalues of their associated self-adjoint matrices. Both questions are addressed in this comment and are answered positively. A general formula for the evolution in every step of mentioned stochastic linear approximation to DRM is found as well a map from a binomial expansion of these matrices key products to a well described fractal object.