Asymptotic sets in networks of coupled quadratic nodes

We study asymptotic behaviour in networks of n nodes with discrete quadratic dynamics. While single map complex quadratic iterations have been studied over the past century, considering ensembles of such functions, organized as coupled nodes in an oriented network, generates new questions with potentially interesting applications to the life sciences. We discuss extensions of traditional results from single map iterations, such as the existence of an escape radius; we then investigate whether crucial information about the network is encoded in the behaviour of the critical orbit. We use two previously defined objects: the network Mandelbrot set (i.e., the set of quadratic parameters in C-n for which the network is post-critically bounded in C-n) and the equi-M set (the diagonal slice of the network Mandelbrot set, corresponding to all nodes using the identical quadratic map). Using a combination of analytical techniques and numerical simulations, we study topological properties of the equi-M set, with the aim of understanding which of these properties are affected by altering different aspects of the network architecture and node-to-node connectivity strengths. We find that, while equi-M sets no longer have a hyperbolic bulb structure, some of their geometric landmarks (e.g., the cusp) are preserved for any network configuration, and other properties (such as connectedness) depend on the network structure. We further study the relationship between the Mandelbrot set and the connectedness locus of the network uni-Julia set (defined as the set of z(0) is an element of C for which all nodes remain bounded when they are all initialed identically at z(0)). We discuss using the geometry of uni-Julia or equi-M sets to classify asymptotic behaviour in networks based on their underlying graph structure. Finally, we propose using a form of averaging uni-Julia and equi-M sets to describe statistically the likelihood of a specific asymptotic behaviour, considered over an entire collection of configurations. We discuss which analytical results can be further supported or refined in the future. We also revisit the ties with applications to the life sciences. We explore how this theoretical study may inform on using similar methods to understand natural systems with more complex architecture and node-wise dynamics.