# An efficient implementation of the Hill-Harmonic Balance method to obtain Floquet exponents and solutions for homogeneous linear periodic differential equations

## TL;DR

This paper presents an efficient Fourier-based implementation of the Hill-Harmonic Balance method to accurately compute Floquet exponents and solutions for homogeneous linear periodic differential equations, verified against known solutions.

## Contribution

The paper introduces a high-precision, Fourier analysis-based implementation of the Hill-Harmonic Balance method utilizing a variational principle to determine Floquet exponents.

## Key findings

- High accuracy in computing Floquet exponents demonstrated on test systems
- Explicit approximate solutions derived from Floquet exponents
- Method effectively handles systems with known and unknown solutions

## Abstract

We propose an implementation of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This implementation uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.