HarmonicBalance.jl is a Julia package for solving periodic, 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.

Example: steady states of a nonlinear resonator

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

using HarmonicBalance, Plots
@variables α ω ω0 F η t 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)

# solve for a range of ω
result = get_steady_states(harmonic_eq, ω => range(0.9, 1.2, 100), (α => 1., ω0 => 1.0, F => 0.01, η => 0.1))

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)

BibTeX entry

@Article{10.21468/SciPostPhysCodeb.6,
    title={{HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance}},
    author={Jan Košata and Javier del Pino and Toni L. Heugel and Oded Zilberberg},
    journal={SciPost Phys. Codebases},
    pages={6},
    year={2022},
    publisher={SciPost},
    doi={10.21468/SciPostPhysCodeb.6},
    url={https://scipost.org/10.21468/SciPostPhysCodeb.6}
}

See also

• JosephsonCircuits.jl: Models superconducting circuits using modified nodal analysis and harmonic balance with an analytic Jacobian.
• Manlab: A similar package in Matlab also using continuation methods and using the Harmonic Balance method for periodic orbits.