Computer-aided analysis of high-dimensional Glass networks: periodicity, chaos, and bifurcations in a ring circuit

Belgacem, Ismail; Edwards, Roderick; Farcot, Etienne

Abstract

Glass networks model systems of variables that interact via sharp switching. A body of theory has been developed over several decades that, in principle, allows rigorous proof of dynamical properties in high dimensions that is not normally feasible in nonlinear dynamical systems. Previous work has, however, used examples of dimensions no higher than 6 to illustrate the methods. Here, we show that the same tools can be applied in dimensions at least as high as 20. An important application of Glass networks is to a recently proposed design of a true random number generator that is based on an intrinsically chaotic electronic circuit. In order for analysis to be meaningful for the application, the dimension must be at least 20. Bifurcation diagrams show what appear to be periodic and chaotic bands. Here, we demonstrate that the analytic tools for Glass networks can be used to rigorously show where periodic orbits are lost and the types of bifurcations that occur there. The main tools are linear algebra and the stability theory of Poincaré maps. All main steps can be automated, and we provide computer code. The methods reviewed here have the potential for many other applications involving sharply switching interactions, such as artificial neural networks.

Citation

Belgacem, I., Edwards, R., & Farcot, E. (2025). Computer-aided analysis of high-dimensional Glass networks: periodicity, chaos, and bifurcations in a ring circuit. Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(5), Article 053151. https://doi.org/10.1063/5.0243955

• Journal Article Type: Article
• Acceptance Date: Jan 29, 2025
• Online Publication Date: May 20, 2025
• Publication Date: May 20, 2025
• Deposit Date: Apr 1, 2025
• Publicly Available Date: Apr 2, 2025
• Journal: Chaos: An Interdisciplinary Journal of Nonlinear Science
• Print ISSN: 1054-1500
• Electronic ISSN: 1089-7682
• Publisher: American Institute of Physics
• Peer Reviewed: Peer Reviewed
• Volume: 35
• Issue: 5
• Article Number: 053151
• DOI: https://doi.org/10.1063/5.0243955
• Keywords: Nonlinear systems, Chaotic bands, Dynamical systems, Stability theory, Electronic circuits, Artificial neural networks, Algebra, Phase space methods
• Public URL: https://nottingham-repository.worktribe.com/output/47276452
• Publisher URL: https://pubs.aip.org/aip/cha/article/35/5/053151/3347409/Computer-aided-analysis-of-high-dimensional-Glass
• Additional Information: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Belgacem, I., Edwards, R., & Farcot, E. (2025). Computer-aided analysis of high-dimensional Glass networks: periodicity, chaos, and bifurcations in a ring circuit. Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(5), Article 053151 and may be found at https://pubs.aip.org/aip/cha/article/35/5/053151/3347409/Computer-aided-analysis-of-high-dimensional-Glass