Complex-Dynamic Fractality and Its Applications in Life Sciences
important for life-science applications because it possesses the essential living system
properties absent in any unitary model, including autonomous dynamic adaptability,
“purposeful” self-development, intrinsic mixture of omnipresent randomness with often
implicit but strong order, and the resulting qualitatively superior dynamic efficiency.
These properties are unified within the universal dynamic symmetry, or conservation, of
complexity [9,11,12] providing the general framework for the described process of
interaction development into a probabilistically fractal structure. The initial interaction
configuration, as described by the starting equations (1), (2), (5), is characterised by the
latent, “potential” complexity form of dynamic information, universally measured by
generalised action. System structure emergence in the form of unreduced dynamical
fractal, eqs. (8)-(22), is described by unceasing transformation of dynamic information
into a dual complexity form, dynamic entropy, generalising the usual entropy to any real
system dynamics and reflecting the fully developed structure. Symmetry of complexity
means that the sum of dynamic information and entropy, or total complexity, remains
unchanged for any given system or process, which gives rise to the universal Hamilton-
Schrodinger formalism mentioned above and extended, causally complete versions of
all other (correct) laws and principles. Due to the intrinsic randomness of the unreduced
fractality and contrary to any unitary symmetry, the universal symmetry of complexity
relates irregular, configurationally “asymmetric” structures and elements, while
remaining always exact (unbroken), which is especially important for description of
biological, explicitly irregular, but internally ordered structures. Constituting thus the
unreduced symmetry of natural structures, the symmetry of complexity extends
somewhat too regular symmetry of usual fractals and approaches the fractal paradigm to
the unreduced complexity of living organism structure and dynamics.
3 Exponentially high efficiency of unreduced fractal dynamics
The probabilistic dynamical fractal, eqs. (8)-(22), emerges as a single whole,
which means that the fractal hierarchy of realisations appears and adapts its structure in
a “real-time” period, comparable with the time of structure formation of the first level of
fractality. This is the complex-dynamical, multivalued, genuine parallelism of real
system dynamics absent in unitary models that try to imitate it by artificial division of
sequential thread of events between simultaneously working multiple units of
interaction, which can be useful, but does not provide any true gain in power. By
contrast, the real, exponential power increase is obtained in natural systems with many
interacting units at the expense of irreducible dynamic randomness, which constitutes
the necessary, but actually quite advantageous “payment” for the huge power growth of
creative interaction processes (whereas any unitary, regular dynamics is strictly
deprived of genuine creativity).
System operation power P is proportional to the number of realisations emerging
within a given time interval, i.e. to the unreduced dynamic complexity: P = P0 C (Nə,
where P0 is a coefficient conveniently taken to be equal to the corresponding unitary
power value (dynamically single-valued, sequential operation model, or “generalised
Turing machine”). Then the relative growth of complex-dynamical fractal power with
respect to unitary model, δP , is given by the unreduced system complexity, which can
be estimated by the fractal realisation number: δP = P/P0 = C ( Nə = Nw-1 ≡ N*
(Nя »1). According to the analysis of section 2, we have the complex-dynamical
fractal hierarchy of system realisations with NQ levels, each of them producing a new