Development of the method of computer analogy for studying and solving complex nonlinear systems

TL;DR

This paper introduces a novel method based on computer analogy for solving complex nonlinear systems of differential equations, including the Lorenz system, by decomposing solutions into regular and random parts and applying filtering techniques.

Contribution

It extends the series solution method with a new reduction procedure and shifting filter, enabling qualitative analysis and approximate solutions of complex nonlinear ODE systems.

Findings

Decomposition into regular and random parts improves solution analysis.

The shifting procedure acts as an effective filter for the random component.

Approximate solutions can provide qualitative insights even without full convergence.

Abstract

A method of representation of a solution as segments of the series in powers of the step of the independent variable is expanded for solving complex systems of ordinary differential equations (ODE): the Lorenz system and other systems. A new procedure of reduction of the representation of the solution to a sum of two parts (regular and random) is performed. A shifting procedure is applied in each level of the independent variable to the random part and it acts as the filter that extracts the values to the regular part. In certain cases it is possible to omit the random part and construct the approximation which does not converge but still provides the qualitative information about the full solution (a linear approximation provides a simple exact solution). Evaluation of the error for this case is performed. Constructing the analytical representation of the solutions for these systems by the developed method is presented.