Transport Catastrophe Analysis as an Alternative to a Monofractal Description: Theory and Application to Financial Crisis Time Series

The goal of this investigation was to overcome limitations of a persistency analysis, introduced by Benoit Mandelbrot for monofractal Brownian processes: nondifferentiability, Brownian nature of process, and a linear memory measure. We have extended a sense of a Hurst factor by consideration of a phase diffusion power law. It was shown that precatastrophic stabilization as an indicator of bifurcation leads to a new minimum of momentary phase diffusion, while bifurcation causes an increase of the momentary transport. An efficiency of a diffusive analysis has been experimentally compared to the Reynolds stability model application. An extended Reynolds parameter has been introduced as an indicator of phase transition. A combination of diffusive and Reynolds analyses has been applied for a description of a time series of Dow Jones Industrial weekly prices for the world financial crisis of 2007–2009. Diffusive and Reynolds parameters showed extreme values in October 2008 when a mortgage crisis was fixed. A combined R/D description allowed distinguishing of market evolution short-memory and long-memory shifts. It was stated that a systematic large scale failure of a financial system has begun in October 2008 and started fading in February 2009. 1. Introduction In 1955, the American researcher Hassler Whitney has created a mathematical foundation of a modern catastrophe theory—the theory of mapping singularities [1]. It includes investigations of peculiarity classes that appear for mapping of one two-dimensional surface to another one. Whitney has found out two stable types of mappings—types that have not been destroyed after negligible deformations of surfaces or their projections. These types of mappings have been generalized for arbitrary manifolds with dimensions up to 10 by Whitney’s followers; see, for example, [2]. One of them led to the discrete change of a system’s characteristic state—“cusp” catastrophe. It is represented for a one-dimensional case in Figure 1. The multiplicity or uncertainty is maximal in the unstable area of C-vicinity. According to [1], the disruption appears as a fusion of stable and unstable regimes, marked by ovals. In terms of a bifurcation theory, this one-dimensional evolution corresponds to the saddle-node fusion in a phase space. Another type of destabilization is a self-oscillating destabilization, suggested by H. Poincare (1879) in his dissertation thesis. It has been proved by A. Andronov and E. Leontovich in 1939. Figure 1: Appearance of Whitney’s “cusp.” According to the Andronov-Leontovich theorem [3], a birth of a