https://github.com/quantumengineeredsystems/harmonicbalance.jl

A Julia package for solving nonlinear differential equations using the harmonic balance method.
https://github.com/quantumengineeredsystems/harmonicbalance.jl

floquet floquet-theory harmonic-balance julia nonlinear-dynamics oscillators perturbation-theory symbolic-computation time-dependent-hamiltonian time-dependent-systems

Last synced: 8 months ago
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A Julia package for solving nonlinear differential equations using the harmonic balance method.

• Host: GitHub
• URL: https://github.com/quantumengineeredsystems/harmonicbalance.jl
• Owner: QuantumEngineeredSystems
• License: bsd-3-clause
• Created: 2021-12-06T16:15:14.000Z (over 4 years ago)
• Default Branch: master
• Last Pushed: 2025-10-20T11:48:35.000Z (8 months ago)
• Last Synced: 2025-10-20T13:28:44.520Z (8 months ago)
• Topics: floquet, floquet-theory, harmonic-balance, julia, nonlinear-dynamics, oscillators, perturbation-theory, symbolic-computation, time-dependent-hamiltonian, time-dependent-systems
• Language: Julia
• Homepage: https://quantumengineeredsystems.github.io/HarmonicBalance.jl/stable/
• Size: 67.4 MB
• Stars: 79
• Watchers: 2
• Forks: 18
• Open Issues: 15
• Metadata Files:
  • Readme: README.md
  • License: LICENSE.md
  • Citation: CITATION.bib

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**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,

```julia

using Pkg

Pkg.add("HarmonicBalance")

```

## Documentation

For a detailed description of the package and examples, see the [documentation](https://quantumengineeredsystems.github.io/HarmonicBalance.jl).

[This repo](https://github.com/quantumengineeredsystems/HarmonicBalance-notebooks) contains a collection of example notebooks.

## [Example: steady states of a nonlinear resonator](https://quantumengineeredsystems.github.io/HarmonicBalance.jl/dev/tutorials/steady_states)

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



```julia

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

```

```julia

plot(result, "sqrt(u1^2 + v1^2)")

```



## Citation

If you use HarmonicBalance.jl in your project, we kindly ask you to cite [this paper](https://scipost.org/SciPostPhysCodeb.6), 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

```bibtex

@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](https://github.com/kpobrien/JosephsonCircuits.jl): Models superconducting circuits using modified nodal analysis and harmonic balance with an analytic Jacobian.

- [Manlab](https://manlab.lma.cnrs-mrs.fr/spip/): A similar package in Matlab also using continuation methods and using the Harmonic Balance method for periodic orbits.