The harmonic balance method for finding approximate periodic solutions of the Lorenz system

TL;DR

This paper applies the harmonic balance method to approximate periodic solutions of the Lorenz system, developing a general symbolic approach and validating results with high-precision numerical methods.

Contribution

It introduces a general symbolic formulation for the harmonic balance method applied to the Lorenz system and demonstrates its effectiveness through computational experiments.

Findings

Coefficients of trigonometric polynomials approximating solutions

Initial conditions and cycle periods obtained

Validation with high-precision numerical integration

Abstract

We consider the harmonic balance method for finding approximate periodic solutions of the Lorenz system. When developing software that implements the described method, the math package Maxima was chosen. The drawbacks of symbolic calculations for obtaining a system of nonlinear algebraic equations with respect to the cyclic frequency, free terms and amplitudes of the harmonics, that make up the desired solution, are shown. To speed up the calculations, this system was obtained in a general form for the first time. The results of the computational experiment are given: the coefficients of trigonometric polynomials approximating the found periodic solution, the initial condition, and the cycle period. The results obtained were verified using a high-precision method of numerical integration based on the power series method and described earlier in the articles of the authors.