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

A novel approach to causal functionals is presented. Combinatorial interpretations of the solutions of nonlinear differential equations with forcing terms are introduced. This theory parallels the algebraic approach with formal power series in noncommutative variables. It makes use of certain combinatorial objects called weighted increasing trees, weighted paths, and histories. Very efficient algorithms for the computation of the corresponding Volterra kernels can be deduced. An introduction to the combinatorial theory is given. An example with a nonlinear circuit is included