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Complex-Dynamical Extension of the Fractal
Paradigm and Its Applications in Life Sciences*
A.P. Kirilyuk
Institute of Metal Physics, 36 Vernadsky Av, Kiev-142, 03142 Ukraine
e-mail: [email protected]
Summary. Complex-dynamical fractal is a hierarchy of permanently, chaotically changing versions of
system structure, obtained as the unreduced, causally probabilistic general solution to an arbitrary
interaction problem. Intrinsic creativity of this extension of usual fractality determines its exponentially
high operation efficiency, which underlies many specific functions of living systems, such as autonomous
adaptability, “purposeful” development, intelligence and consciousness (at higher complexity levels). We
outline in more detail genetic applications of complex-dynamic fractality, demonstrate the dominating
role of genome interactions, and show that further progressive development of genetic research, as well as
other life-science applications, should be based on the dynamically fractal structure analysis of interaction
processes involved. We finally summarise the obtained extension of mathematical concepts and
approaches closely related to their biological applications.
1 Introduction
The success of fractal paradigm in bio-system structure analysis, as presented in this
series of conferences [1-3], reflects high efficiency of fractal geometry in life function
realisation conceived and used by nature itself. In a broader sense, fractal structure
efficiency appears inevitably and naturally in a wide variety of real processes, from
physico-chemical structures to economic system evolution [4-8], driven by unreduced
interaction processes and often referred to as systems with complex dynamics. Using
the universally nonperturbative analysis of a generic interaction process, we have
rigorously specified the connection between fractality and dynamic complexity [9,10],
where the extended, complex-dynamic fractality has been derived as inevitably
emerging structure of any real interaction process. In that way, the dynamic complexity
as such acquires a rigorous and universally applicable definition, while the fractal
structure of a real interaction is obtained as the truly complete, dynamically multivalued
(probabilistic) general solution of a problem, replacing its reduced, dynamically single-
valued (regular) version. The dynamically probabilistic, permanently changing fractal of
real system dynamics is a natural extension of the canonical, “geometric” fractality
possessing an involved, but basically predictable (regular) and fixed structure.
Complex-dynamic fractality is not a “model” any more, but the unreduced version of
any real, “nonintegrable” and “nonseparable” system structure and dynamics, which is
especially interesting for fractality involvement with living systems because it provides
rigorously derived versions of those essential life properties — such as intrinsic
adaptability, self-development and “reasonable” behaviour — that determine its specific
efficiency and remain largely “mysterious” within usual, perturbative theory.
In this report, after recalling the mathematical framework of complex-dynamic
fractality (section 2), we proceed to further exploration of its properties important for
life-science applications. We show that due to the hierarchy of unceasing probabilistic
change of the living fractal structure, its power to perform useful functions grows
* Report presented at the IVth International Symposium “Fractals in Biology and Medicine” (Ascona, 10-
13 March 2004), http://www.fractals.issi.cerfim.ch/.