Generating series of a network of interconnected systems

Consider a complex network of interconnected input-output systems. The input can, for example, be a time-series or a curve in $\mathbb{R}^m$. The objective is to understand the resulting input-output system represented by the network.

This is accomplished by providing an extension of the notion of a "Kawski-Sussmann formal system". This is a formal ODE whose solution is viewed as a trajectory evolving in an infinite-dimensional formal Lie group $G$ and whose tangent space at each point is identified with the shuffle algebra over $X$. It is shown that this method can produce the generating series of any of the networks appearing in the literature, but it can also be applied to networks that are too complex for any existing method. (Based on joint work with W. S. Gray.)