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

by Jan Košata, Javier del Pino, Toni L. Heugel, Oded Zilberberg

This is not the latest submitted version.

Submission summary

As Contributors: Jan Košata · Javier del Pino
Preprint link: scipost_202202_00005v1
Code repository:
Date submitted: 2022-02-03 16:26
Submitted by: del Pino, Javier
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


HarmonicBalance.jl is a publicly available Julia package designed to simplify and solve sys- tems 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 ex- ploration 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

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2022-4-4 (Invited Report)


- The article is well written. It is a good pedagogical introduction to the topic of nonlinear resonators.

- It represents a good manual for the potential users of the software package.


- The literature about the topic is not complete.

- At conceptual level, it does not introduce any novelty. The package performs standard numerical methods and approximations that are commonly used by the experts in the field.

- The target system is also not clear. It is not clear what are the scientific motivations and research interests in investigating a large number of N coupled resonators instead of a few of them (N=1, 2 ,3) or many harmonic components of a single resonator.


(see attached file)


  • validity: high
  • significance: low
  • originality: low
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Author:  Javier del Pino  on 2022-05-16  [id 2476]

(in reply to Report 2 on 2022-04-04)

Response can be found in the attached .pdf.



Anonymous on 2022-05-23  [id 2504]

(in reply to Javier del Pino on 2022-05-16 [id 2476])

I read the reply and the authors satisfactorily answered to most of my criticism.
In a first reading, I had an oversight on some functionalities already implemented in the code.
I am still a bit sceptic about the relevance for experimental systems with an increasing number of resonator modes but I agree that the work can be published on SciPost Codebases.

Anonymous Report 1 on 2022-3-21 (Invited Report)


1 - Very well written, provides a very clear review of the relevant subjects.
2 - The presented software package is very useful


1 - Scientific novelty is quite low, package does not introduce any new tools or apply them in a novel manner.


See attached file.


  • validity: high
  • significance: ok
  • originality: low
  • clarity: top
  • formatting: excellent
  • grammar: excellent

Author:  Javier del Pino  on 2022-05-17  [id 2480]

(in reply to Report 1 on 2022-03-21)

Response can be found in attached .pdf.



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