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

TL;DR

This paper extends analytical methods for high-dimensional Glass networks, demonstrating their application to systems with at least 20 dimensions, including chaotic electronic circuits, and providing automated tools for bifurcation analysis.

Contribution

It shows that existing theoretical tools can be applied to analyze high-dimensional Glass networks, including chaotic systems, with automated computational support.

Findings

Bifurcation diagrams reveal periodic and chaotic regimes.

Analytic methods can rigorously identify bifurcation points.

Tools are automated and applicable to other systems with switching interactions.

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 dimension 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 Poincar\'e 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.