HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance

TL;DR

HarmonicBalance.jl is a Julia package that efficiently solves nonlinear periodic systems using harmonic balance, symbolic algebra, and homotopy continuation, with a user-friendly interface for exploring steady-state behaviors.

Contribution

The paper introduces HarmonicBalance.jl, a Julia package that combines harmonic balance, symbolic algebra, and homotopy continuation to analyze nonlinear dynamics with a graphical interface.

Findings

Successfully computes all steady-state solutions.

Enables exploration of complex steady-state topologies.

Provides a user-friendly GUI for phase diagrams.

Abstract

HarmonicBalance.jl is a publicly available Julia package designed to simplify and solve systems of periodic time-dependent nonlinear ordinary differential equations. Time dependence of the system parameters is treated with the harmonic balance method, which approximates the system's behaviour as a set of harmonic terms with slowly-varying amplitudes. Under this approximation, the set of all possible steady-state responses follows from the solution of a polynomial system. In HarmonicBalance.jl, we combine harmonic balance with contemporary implementations of symbolic algebra and the homotopy continuation method to numerically determine all steady-state solutions and their associated fluctuation dynamics. For the exploration of involved steady-state topologies, we provide a simple graphical user interface, allowing for arbitrary solution observables and phase diagrams. HarmonicBalance.jl is a free software available at https://github.com/NonlinearOscillations/HarmonicBalance.jl.