Chapter 6 Diverse Applications In this final chapter, we turn our attention to more practical issues, some of them concerned with the analysis of real data. Hitherto we have been interested in under- standing the origin and nature of chaos as it occurs in known, deterministic systems. We can see the system is chaotic, but we know what the governing equations are, and the process is fully described by them. In applications, we sometimes assume that the system is deterministic, even if we are unable to analyse the equations. For example, we think of turbulence in fluids as representing the chaotic behaviour of solutions of the Navier–Stokes equations, although at best this is only an inference. In other situations, we may conceive of the system behaviour as having a certain structured randomness. An example of this is the stock market, which is usually thought of as a directed random walk. The question then arises, how do we deal with such systems in practice? 6.1 Chaotic Data As a taster, consider Fig. 6.1. It shows three different time series. The first of these is a solution of the Mackey–Glass equation, a delay-recruitment equation of the form ε ˙ x =−x + f (x 1 ), (6.1) where x 1 = x (t − 1) and ε ≪ 1. The function f (x ) = λx (1 − x ) has the typical unimodal shape, which induces chaos in one-dimensional maps (when ε = 0 in (6.1)), and when ε is small, it is known that trajectories are chaotic. The second series in Fig. 6.1 is a realisation of a stochastic process consisting of what is known as red noise 1 . While bearing a superficial resemblance to the first series, it is (at least, partly) determined by random events. The final series represents actual data, being 1 Noise will be discussed below. © Springer Nature Switzerland AG 2019 A. Fowler and M. McGuinness, Chaos, https://doi.org/10.1007/978-3-030-32538-1_6 207