Finite-dimensional behaviour and observability in a randomly forced PDE

In earlier work [D.S. Broomhead, J.P. Huke, M.R. Muldoon, and J. Stark, Iterated function system models of digital channels, Proc. R. Soc. Lond. A 460 (2004), pp. 3123-3142], aimed at developing an approach to signal processing that can be applied as well to nonlinear systems as linear ones, we produced mathematical models of digital communications channels that took the form of iterated function systems ( IFS). For finite-dimensional systems these models have observability properties indicating they could be used for signal processing applications. Here we see how far the same approach can be taken towards the modelling of an infinite-dimensional system. The cable equation is a well-known partial differential equation (PDE) model of an imperfectly insulated uniform conductor, coupled to its surroundings by capacitive effects. ( It is also much used as a basic model in theoretical neurobiology.) In this article we study the dynamics of this system when it is subjected to randomly selected discrete input pulses. The resulting IFS has a unique finite-dimensional attractor; we use results of Falconer and Solomyak to investigate the dimension of this attractor, relating it to the physical parameters of the system. Using work of Robinson, we show how some of the observability properties of the IFS model are retained.