A.P. Kirilyuk
split into Nξ = N realisations (where N is the number of system components and NQ is
the number of their operative states). So the total (maximum) realisation number Nw of
the dynamical fractal, and thus also δP , grows exponentially with NQ :
δP ≡ Nw = NnQ . (23)
Truly complicated systems from superior complexity levels, such as genome, cell, or
brain dynamics, have high values of N and NQ , so that their exponential combination of
eq. (23) produces not only quantitative, but also qualitative effects appearing as various
“miracles” of “living” and “intelligent” behaviour that cannot be convincingly imitated
by unitary models (and now we know the exact, fundamental reason for that).
The estimate of eq. (23) refers, however, to a single interaction “run” at a given
level of complexity describing the emergence of one “compound”, fractally structured
realisation of the first level. System structure formation in the process of its operation
does not stop there and involves a hierarchy of interactions at superior levels, where the
above fractal structure within a given level plays the role of distributed “interaction
transmitter” between harder, first-level parts of fractality. This means that the dynamic
fractal grows, starting from a given interaction level, not only “in depth” (to generally
smaller scales and lower complexity sublevels), but also to higher complexity levels. In
order to estimate the total relative efficiency of such systems of “biologically high”
complexity, consider a many-body interaction system consisting of Nunit operative units
(such as neurons, or genes, or relevant cell components) each of them connected by
nlink effective links to other units, so that the total number of interaction links in the
system is N = Nunitnlink. The number of system realisations Nw, and thus δP , is of the
order of the number of all possible combinations of links, Nw - N!, which is the
distinctive feature of the unreduced, dynamically multivalued fractality [11]:
. I------Ґ N I N .r . .
δP = N* - N! - √2∏N INI ~ Nn , (24)
к e )
where we have used the well-known Stirling formula valid for large N (which is greater
than 1012 for both brain and genome interaction structure, see section 4). For the case of
N ~ 1012 the estimate of eq. (24) gives δP » 101013 » 101012 ~ 10n , which is a practical
infinity demonstrating the qualitatively huge efficiency of complex-dynamic fractality
and its causal origin. Note that any unitary (basically regular and sequential) model of
the same system dynamics would give the operation power growing only as Nβ (β ~ 1)
and remaining negligible with respect to exponentially big efficiency of unreduced
complex dynamics (including its unique adaptability and creativity).
4 Causally complete genetics, integral medicine, and other
applications of the unreduced complex-dynamic fractality
Causally complete understanding of complex-dynamical fractal structure development
in real biological and bio-inspired systems leads to a number of promising applications
in life sciences, where modification and control of bio-system dynamics deal with its
realistic, unreduced version and are comparable with natural creation processes. The
relevant examples include (see also [9-11]) (1) causally complete understanding and use