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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

Submission summary

As Contributors: Jan Košata · Javier del Pino
Arxiv Link: (pdf)
Code repository:
Date accepted: 2022-06-02
Date submitted: 2022-05-18 17:38
Submitted by: del Pino, Javier
Submitted to: SciPost Physics Codebases
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 systems 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 exploration 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:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics Codebases: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

Author comments upon resubmission

Dear Dr Attaccalite,

Thank you for sending us the referee reports on our manuscript "HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance". We are pleased to hear that the referees, and you with them, find our work of potential interest. We have addressed the referees' comments in total and used them to improve our manuscript.

We are glad that the Referees recognise our open-source software's critical range of applications and praise the clarity and quality of our presentation. They raise several questions regarding the scope of our toolbox that we each address in detail in our response and revised manuscript. We also appreciate that Referee 1 finds our package very useful and convenient, while they suggest package enhancements such as calculation indicators and data exports on the results. Referee 2 raises comments on the expected impact in several disciplines and the capabilities of the current version of the package. We have followed up on the suggestions, and while this has required significant effort, it has led to several extension modules included in the latest release and new usage examples and documentation. We are delighted with the positive impact it has had on our toolbox, whose scope has now increased.

Finally, both referees highlight the importance of an extension that would allow determining limit cycles and stochastic dynamics. Here, we emphasise that our current submission's goal is to provide a solution to the (so far unsolved problem) of finding the totality of fixed point steady-state solutions o a deterministic nonlinear, time-dependent differential equation. In this process, we also imbue into the code several years of experience in handling the solutions to provide an explainable output beyond the scope of time-dependent-based numerical experiments. Therefore, while we have carried out preliminary steps in the suggested directions, we plan to include limit cycle functionality and stochastic dynamics in full in future releases of the package.

The responses to each of the Referee's replies can be found below each corresponding report. We hope that, with these changes, the referees find our improved manuscript suitable for publication in SciPost Codebases. I thank you for your careful consideration.

With kind regards,

Javier del Pino (on behalf of the authors)

List of changes

* We have included nine new relevant references to nonlinear dynamics in the Introduction. Here we added minor revisions to establish further the different capabilities of our methodology and time evolution toward the steady-state.

* We revised section 2.2 to stress the envisioned usage of harmonic expansions to find limit cycles. Here we included four new references.

* We have extended the comparison with other harmonic balance implementations by further elaborating
the first bullet point in section 4.

* We have revised our plotting routines as detailed above and upgraded our progress bar to offer a
human-readable output during parallel calculations and considering progress along with analysis tasks
(matching solution branches, calculating response spectra).

* To further emphasise the time-domain simulation capabilities of the package, we incorporated in the
manuscript a link to a new relevant example in the online documentation.

* We now provide a LinearResponse module to calculate linear response, as documented in

* We included a new example to illustrate the calculation of a sub-harmonic response in HarmonicBalance.jl, namely a parametrically driven Duffing oscillator or "parametron". Here, we included five new references.

* We added a comment in the revised manuscript regarding the focus on deterministic dynamics.
We shaped the discussion in section 5.4, “Increasing computational complexity: performance scaling.”
by highlighting some example experimental directions where our library can be most appropriate.

*In unison, we stressed the potential impact in other research areas in the conclusions.

Reports on this Submission

Anonymous Report 2 on 2022-5-25 (Invited Report)


- good introduction to the problem
- new numerical codes for solving coupled nonlinear equations


- no novelty at conceptual level


The authors properly replied to my questions.
I have still some doubts on the usefulness of this tool for the case of many resonator modes but I think that this work deserve publication in SciPost Codebases.

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

Anonymous Report 1 on 2022-5-24 (Invited Report)


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


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


The authors have addressed all of my concerns, and I believe the paper deserves publication in SciPost Codebases.

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

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