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Discrete Dynamics in Nature and Society
Volume 7, Issue 1, Pages 23-26
http://dx.doi.org/10.1080/10260220290013480

A dynamic system interpretation of irreducible complexity

Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Que. H4B 1R6, Canada

Received 24 April 2001

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Behe recently defined the idea of irreducible complexity for biological systems. Using the language of mathematics, we reinterpret his definition from a dynamical systems perspective. Our basic premise is that living organisms behave dynamically in a chaotic way while predictable periodic behavior reflects cessation of function. We consider the dynamics of a functioning system and altered versions of it to draw conclusions about the irreducible complexity of the original system. The dynamics of an organism is described by means of a discrete time transformation τ on the phase space of the system. The statistical behavior of τ is studied by means of its Frobenius–Perron operator which, in special cases, can be represented by a matrix. Using these matrices we rewrite our definition of irreducible complexity:M is irreducibly complex if it is primitive but no principal submatrix of M is primitive. The primitivity property implies chaotic behavior, while failure to have the primitivity property reflects periodic behavior. Examples of irreducibly complex dynamical systems are presented. We show that certain dynamical systems which are irreducibly complex have an additional property, namely that other systems arbitrarily close to it behave in a dramatically different way. Such behavior suggests that selective evolution by means of small perturbations may not be a general mechanism for achieving the dynamical behavior of a complex system.