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Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems
by Samuel Cure, Izaak Neri
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Samuel Cure · Izaak Neri |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2112.13498v3 (pdf) |
| Date accepted: | 2023-02-14 |
| Date submitted: | 2022-11-17 07:16 |
| Submitted by: | Cure, Samuel |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We analyse the stability of large, linear dynamical systems of variables that interact through a fully connected random matrix and have inhomogeneous growth rates. We show that in the absence of correlations between the coupling strengths, a system with interactions is always less stable than a system without interactions. Contrarily to the uncorrelated case, interactions that are antagonistic, i.e., characterised by negative correlations, can stabilise linear dynamical systems. In particular, when the strength of the interactions is not too strong, systems with antagonistic interactions are more stable than systems without interactions. These results are obtained with an exact theory for the spectral properties of fully connected random matrices with diagonal disorder.
Author comments upon resubmission
We apologise for the delayed response.
We took the time to carefully address all points raised by the Referees.
In addition, we have included exact analytical results on the leading eigenvalue when the the mean value of the matrix entries is nonzero. This result was missing in the previous manuscript as it involves a different mathematical technique involving outliers, and hence we put, admittedly somewhat artificial, the mean value of the matrix entries to zero. However, in the new version of the manuscript we provide exact analytical results for the eigenvalue outliers when the mean values of the matrix entries are nonzero, and we also discuss the effect of antagonistic interactions on linear stability when the leading eigenvalue is an outlier.
Given these improvements, we hope the paper will be considered favourably for publication in SciPost Physics.
In the reply to the referee, you can find a detailed reply to all the Referee's comments [Referee comments in blue and our reply in black].
Kind regards,
The Authors
List of changes
- Added Section 5;
- Added Appendices E and F;
- Added Figures 5 and 7;
- Clarified derivation steps in Appendix A;
- Included several references;
- Changed the title and abstract following comments of Referee B;
- We have taken the comments of the Referees onboard and made several local changes in the manuscript (see response to referee reports).
Published as SciPost Phys. 14, 093 (2023)