Chaos Theory vs FMI

Realizing a Vision

The quest to use chaos theoretic tools to model time series, in particular financial time series, is many decades old. It is not presumptuous to claim that FMI realizes many of the goals envisioned by chaos theorists of the 90s. This is due to interesting dualities between chaos theories and renormalization group flows that have been discussed for many years but never fully fleshed out. As it turns out, the chaos theoretic notion of finding stable 'attractor' solutions to which many trajectories converge are dual to 'fixed points' that RG flows usually begin and end in. In both cases, the special solutions are characterized by enhanced symmetries. The fractal attractors in dynamical systems obey discrete scaling symmetries, whereas the 'fixed points' in field theory obey continuum versions of these. Chaos theoretic bifurcations, which are common in time series, are realized as phase transitions in FMI. Intermittency is related to phase transitions of a different sort. From this discussion, it might appear that we have simply invented a formalism that mirrors chaos theory. This is not true for two reasons. First, field theoretic methods are much more general and systematic and allow plenty of perturbative expansions that are independent of symmetry enhancements seen close to fixed points. Second, non-perturbative methods allow far greater analytic and numerical control on the nonlinear behavior of field theories than methods in chaos theory. As far as time series is concerned, chaos theoretic tools can be viewed as toy models for field theories.