Information Technology Reference
In-Depth Information
where, given that the fractional operator index is in the interval 1
>α>
1
/
2,
1
/
2
H
>
0
.
(5.78)
Consequently, the process described by the dissipation-free fractional Langevin equa-
tion is anti-persistent.
This anti-persistent behavior of the time series was observed by Peng et al .[ 20 ]for
the differences in time intervals between heartbeats. They interpreted this result, as did a
number of subsequent investigators, in terms of random walks with H
2. However,
we can see from ( 5.76 ) that the fractional Langevin equation without dissipation is an
equally good, or one might say better, description of the underlying dynamics. The
scaling behavior alone cannot distinguish between these two models; what is needed
is the complete statistical distribution of the empirical data rather than just the time-
dependence of one or two moments.
<
1
/
5.3.2
Multiplicative random force
So far we have learned that the dynamics of a conservative dynamical web can be mod-
eled using Hamilton's equations of motion. A dynamical web with many degrees of
freedom can be used to construct a Langevin equation from a Hamiltonian model for
a simple dynamical network coupled to the environment. The equations of motion for
the coupled network are manipulated so as to eliminate the degrees of freedom of the
environment from the dynamical description of the web of interest. Only the initial state
of the environment remains in the Langevin description, where the random nature of
the driving force is inserted through the choice of distribution of the initial states of the
environment. The random driver is typically assumed to be a Wiener process - that is,
to have Gaussian statistics and no memory.
When the web dynamics depends on what occurred earlier, that is, the environment
has memory, the simple Langevin equation is modified. The generalized Langevin equa-
tion takes memory into account through an integral term whose memory kernel is
connected to the autocorrelation of the random force. Both the simple and the gener-
alized Langevin equations are monofractal if the fluctuations are monofractal, so the
web response is a fractal random process if the random force is a fractal random pro-
cess and the dynamics are linear. But the web response can be a fractal random process
even when the random force is a Wiener process, if the web dynamics are fractional.
A different way of modeling the influence of the environment by a random force does
not require the many degrees of freedom used in the traditional approach. Instead the
nonlinear dynamics of the bath become a booster due to chaos, and the back-reaction of
the booster to the web of interest provides the mechanism for dissipation. This picture
of the source of the Langevin equation sidesteps a number of the more subtle statistical
physics issues and suggests a more direct way to model fluctuations in complex social
and biological webs.
While the properties of monofractals are determined by the global Hurst exponent,
there exists a more general class of heterogeneous signals known as multifractals which
are made up of many interwoven subsets with different local Hurst exponents h .The
Search WWH ::




Custom Search