Networks of coupled quadratic nodes

TL;DR

This paper explores the extension of classical complex quadratic dynamics to networks of coupled quadratic nodes, analyzing their asymptotic behavior, geometric properties, and potential applications in life sciences.

Contribution

It generalizes Fatou-Julia theory to networked systems, studying properties of Mandelbrot and Julia sets in high-dimensional parameter spaces and their dependence on network structure.

Findings

Network Mandelbrot sets lack hyperbolic bulbs but retain some geometric features.

Connectedness of sets depends on network topology.

A method for analyzing asymptotic dynamics across multiple networks is proposed.

Abstract

We study asymptotic dynamics in networks of coupled quadratic nodes. While single map complex quadratic iterations have been studied over the past century, considering ensembles of such functions, organized as coupled nodes in a network, generate new questions with potentially interesting applications to the life sciences. We investigate how traditional Fatou-Julia results may generalize in the case of networks. We discuss extensions of concepts like escape radius, Julia and Mandelbrot sets (as parameter loci in $\mathbb{C}^n$, where $n$ is the size of the network). We study topological properties of these asymptotic sets and of their two-dimensional slices in $\mathbb{C}$ (defined in previous work). We find that, while network Mandelbrot sets no longer have a hyperbolic bulb structure, some of their geometric landmarks are preserved (e.g., the cusp always survives), and other properties (such as connectedness) depend on the network structure. We investigate possible extensions of the relationship between the Mandelbrot set and the Julia set connectedness loci in the case of network dynamics. We discuss possible classifications of asymptotic behavior in networks based on their underlying graph structure, using the geometry of Julia or Mandelbrot sets as a classifier. Finally, we propose a method for book-keeping asymptotic dynamics simultaneously over many networks with a common graph-theoretical property. \emph{Core} Julia and Mandelbrot sets describe statistically average asymptotic behavior of orbits over an entire collection of configurations.