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Dynamical Systems and Biological Regulations

  • Chapter
Complex Systems

Part of the book series: Nonlinear Phenomena and Complex Systems ((NOPH,volume 6))

Abstract

We consider in this tutorial particular features of many definitions of attractors given in the literature by looking at examples of attracting cycles. We prove by considering specific counterexamples, that there is in fact no order between definitions, only some of them being weaker than others. The chaotic case is also discussed. After which we study mathematical properties of the “natural measure” of a semi-orbit and show that this set function is not a measure. In spite of this fact, this definition may be used in order to define fractal dimensions and dimension functions of an attractor. We compare these dimensions, by studying for example the equality between the two dimension functions proposed by Grassberger, and by Hentschel and Procaccia. Then we study the distribution of the point-wise dimension in the special case of the Baker transformation. We propose one example of application in neuromodelling. We consider after the case of a universal minimal regulatory system (called a regulon), having a positive and a negative circuit in its interaction matrix, and we recall the main results related to the presence of such circuits. Finally, we give two examples of application of the interaction matrix: one concerns the dual problems of synchronization and de-synchronization of a neural model (susceptible to serve as a sketch for hippocampus memory evocation processes), and the second deals with the problem of occurrence of weak parts along the chromosomes, related to the ubiquitory genes expression.

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Author information

Authors and Affiliations

  1. Faculty of Medicine, TIMC-IMAG-CNRS UJF-INPG, 38706, La Tronche, France

    Jacques Demongeot, Julio Aracena, Samia Ben Lamine, Sylvain Meignen & Arnaud Tonnelier

  2. Laboratoire de Génétique, Université Libre de Bruxelles, 67 rue des chevaux, 1640, Rhode Saint Genèse, Belgique

    Rene Thomas

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  1. Jacques Demongeot
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  2. Julio Aracena
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  3. Samia Ben Lamine
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  4. Sylvain Meignen
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  5. Arnaud Tonnelier
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  6. Rene Thomas
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Editors and Affiliations

  1. Department of Mathematical Engineering, Faculty of Physical and Mathematical Sciences, University of Chile, Santiago, Chile

    Eric Goles  & Servet Martínez  & 

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Demongeot, J., Aracena, J., Lamine, S.B., Meignen, S., Tonnelier, A., Thomas, R. (2001). Dynamical Systems and Biological Regulations. In: Goles, E., Martínez, S. (eds) Complex Systems. Nonlinear Phenomena and Complex Systems, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0920-1_3

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  • DOI: https://doi.org/10.1007/978-94-010-0920-1_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-3817-1

  • Online ISBN: 978-94-010-0920-1

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