HarmonicBalance.jl is a Julia package for solving nonlinear differential equations using the method of harmonic balance.
Installation
To install HarmonicBalance.jl, you can use the github repo or the Julia package manager,
using Pkg
Pkg.add("HarmonicBalance")
Documentation
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(ω*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 = (α => 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)")
Citation
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)