A COMBINATORIAL APPROACH TO NONLINEAR FUNCTIONAL EXPANSIONS
Pierre Leroux, Xavier Gérard Viennot, UniversitC du QuCbec, B MontrCal · 1988
In 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-Laganigue. This theory make 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 kemels. In this paper, we just give an introduction to our combinatorial theory. An example with a nonlinear circuit will give a flavor of our approach.The complete proofs and general theory will be exposed in another paper.