A combinatorial approach to nonlinear functional expansions: an introduction with an example

AbstractIn this paper we present a new approach to causal functionals. We introduce combinatorial interpretations of the solutions of nonlinear differential equations with forcing terms. This theory parallels the algebraic approach with formal power series in noncommutative variables developed by Fliess, Lamnabhi and Lamnabhi-Lagarrigue. This theory makes use of certain combinatorial objects called weighted increasing trees, weighted paths and histories. We can deduce very efficient algorithms for the computation of the corresponding Volterra kernels. We present an introduction to our combinatorial theory. An example with a nonlinear circuit gives the flavor of our approach. The complete proofs and general theory will be discussed in a further paper