A Julia package for solving nonlinear differential equations using the harmonic balance method.
Author NonlinearOscillations
30 Stars
Updated Last
1 Year Ago
Started In
December 2021

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HarmonicBalance.jl is a Julia package for solving nonlinear differential equations using the method of harmonic balance.


To install HarmonicBalance.jl, you can use the github repo or the Julia package manager,

using Pkg


For a detailed description of the package and examples, see the documentation.

This repo contains a collection of example notebooks.

Simple example

Let's find the steady states of a driven Duffing oscillator with nonlinear damping, its equation of motion is:

using HarmonicBalance
@variables α, ω, ω0, F, t, η, x(t) # declare constant variables and a function x(t)
diff_eq = DifferentialEquation(d(x,t,2) + ω0^2*x + α*x^3 + η*d(x,t)*x^2 ~ F*cos*t), x)
add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)

# implement ansatz to get harmonic equations
harmonic_eq = get_harmonic_equations(diff_eq)

fixed ==> 1., ω0 => 1.0, F => 0.01, η=>0.1)   # fixed parameters
varied = ω => LinRange(0.9, 1.2, 100)           # range of parameter values
result = get_steady_states(harmonic_eq, varied, fixed)
A steady state result for 100 parameter points

Solution branches:   3
   of which real:    3
   of which stable:  2

Classes: stable, physical, Hopf, binary_labels
plot(result, "sqrt(u1^2 + v1^2)")


If you use HarmonicBalance.jl in your project, we kindly ask you to cite this paper, namely:

HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance Jan Košata, Javier del Pino, Toni L. Heugel, Oded Zilberberg SciPost Phys. Codebases 6 (2022)