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 Zap. Nauchn. Sem. POMI, 2008, Volume 360, Pages 31–69 (Mi znsl2158)

Instability, complexity, and evolution

S. Vakulenkoab, D. Grigorievc

a Institute of Problems of Mechanical Engineering, Russian Academy of Sciences
b North Western Institute of Printing, St. Petersburg State University of Technology and Design
c Université de Lille

Abstract: In this paper, we consider a new class of random dynamical systems which contains in particular neural networks and complicated circuits. For these systems we consider the viability problem: we suppose that the system survives only if the system state is in a prescribed domain $\Pi$ of a phase space. The approach developed here is based on some fundamental ideas proposed by A. Kolmogorov, R. Thom, M. Gromov, L. Valiant, L. Van Valen, and others.
Under some conditions it is shown that almost all systems from this class with fixed parameters are unstable in the following sense: the probability $P_t$ to leave $\Pi$ within time interval $[0,t]$ tends to 1 as $t\to\infty$. However, if it is allowed to change these parameters sometimes (“evolutionary” case), then possibly that $P_t<1-\delta<1$ for all $t$ (“stable evolution”). Furthermore we study the properties of such stable evolution assuming that the system parameters are coded by a dicsrete code. This allows us to apply the complexity theory, coding, algorithms etc. Evolution is a Markov process of this code modification. Under some conditions we show that the stable evolution of unstable systems possesses such general fundamental property: the relative Kolmogorov complexity of the code cannot be bounded by a constant as time $t\to\infty$. For circuit models we define complexity characteristics of these circuits. We find that these complexities also have a tendency to increase during stable evolution. We give concrete examples of stable evolution. Bibl. – 80 titles.

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English version:
Journal of Mathematical Sciences (New York), 2009, 158:6, 787–808

UDC: 517.958:57
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Citation: S. Vakulenko, D. Grigoriev, “Instability, complexity, and evolution”, Representation theory, dynamics systems, combinatorial methods. Part XVI, Zap. Nauchn. Sem. POMI, 360, POMI, St. Petersburg, 2008, 31–69; J. Math. Sci. (N. Y.), 158:6 (2009), 787–808

Citation in format AMSBIB
\Bibitem{VakGri08} \by S.~Vakulenko, D.~Grigoriev \paper Instability, complexity, and evolution \inbook Representation theory, dynamics systems, combinatorial methods. Part~XVI \serial Zap. Nauchn. Sem. POMI \yr 2008 \vol 360 \pages 31--69 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl2158} \elib{http://elibrary.ru/item.asp?id=13759285} \transl \jour J. Math. Sci. (N. Y.) \yr 2009 \vol 158 \issue 6 \pages 787--808 \crossref{https://doi.org/10.1007/s10958-009-9412-4} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-67349178228} 

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