  The goal of this investigation was to overcome limitations of a persistency
analysis, introduced by Benoit Mandelbrot for fractal 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 pre-catastrophic stabilization as an
indicator of bifurcation leads to a new minimum of momentary phase diffusion,
while bifurcation causes an increase of the momentary transport. Basic
conclusions of a diffusive analysis have been compared to the Lyapunov
stability model. An extended Reynolds parameter has been introduces as an
indicator of phase transition. A combination of diffusive and Reynolds analysis
has been applied for a description of a time series of Dow Jones Industrial
weekly prices for a world financial crisis of 2007-2009. Diffusive and Reynolds
parameters shown an extreme values in October 2008 when a mortgage crisis was
fixed. A combined R/D description allowed distinguishing of short-memory and
long memory shifts of a market evolution. It was stated that a systematic large
scale failure of a financial system has begun in October 2008 and started
fading in February 2009.
