The standard approach to modelling a non-linear dynamic system is via a set of differential equations where the rate of change of states is described and an equation, the observation, which is a measure of the system, is described as a function of the current value of the states.

But there are other ways to represent a non-linear system. A classic example is the Volterra series which described the output, the observation, as a function of the linear, second and higher order functions of the temporal history of the input(s) - the so called Volterra kernels.

Two other ways, also series, the Chen-Fliess series and Path Signatures, can also be used to describe the output as a function of the inputs. These representations are in the form of iterated integrals of the past inputs (and maybe time) of the system.

A tutorial on this approach can be found here.

The tutorial shows how individual windows of a non-linear system can be described as Chen-Fliess and Pathe Signature series and then as sets of bilinear ODEs, automata.

Here I show how these series, or automata, can be stitched together using layered automata to reconstruct a system across time windows in much the same way as state-space representations are used.