Basic Results for the Behaviour of Discrete Iterations

  • François Robert
Conference paper
Part of the NATO ASI Series book series (volume 20)


There has been an improved effort for about 20 years in studying the dynamical behaviour of discrete, iterative systems. The reason for this is probably that different (but conceptually similar) discrete models are presently of interest in various domains of science, such as: Physics (spin glass problems, see (10), (12), (14), for example), Chemistry (diffusion reactions (9)), Biomathematics (neural networks, genetic nets, (10), (11), (14), (18), (20)), Computer Science (pattern recognition, associative memories (7), (10), cellular automata (1), (2), (9), (19), (21), (22), (23), cellular arrays for systolic computation in V.L.S.I. systems (13), (15), (17), (19) and so on: see especially (4), (5), (6)).


Cellular Automaton Connectivity Graph Incidence Matrix Systolic Array Serial Mode 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (1).
    A.W. BURKS: Essays on cellular Automata., University of Illinois Press (1970).zbMATHGoogle Scholar
  2. (2).
    E.F. CODD: Cellular Automata., Academic Press (1968).zbMATHGoogle Scholar
  3. (3).
    M. COSNARD, E. GOLES: Dynamique d’un automate à mémoire modélisant le fonctionnement d’un neurone., C.R.A.S., t. 299, I N° 10, (1984). (see also the papers by M. Cosnard and E. Goles in the present volume)Google Scholar
  4. (4).
    M. COSNARD, J. DEMONGEOT, A. LEBRETON, Editors: Rythms in biology and other fields of Applications. Springer Verlag, Lecture notes in Mathematics n°49 (1983).Google Scholar
  5. (5).
    J. DELLA DORA, J. DEMONGEOT, B. LACOLLE, Editors: Numerical methods in study of critical phenomena., Springer Verlag (1981).zbMATHGoogle Scholar
  6. (6).
    J. DEMONGEOT, E. GOLES, M. TCHUENTE, Editors: Dynamic behaviour of automata networks, Academic Press (1984).Google Scholar
  7. (7).
    F. FOGELMAN: Contributions à une théorie du calcul sur réseaux. Thesis, Grenoble 1985. (see also the paper by F. Fogelman in the present volume.)Google Scholar
  8. (8).
    E. GOLES: Comportement dynamique de réseaux d’automates. Thesis, Grenoble 1985. (see also the paper by E. Goles in the present volume.)Google Scholar
  9. (9).
    J.M. GREENBERG, B.D. HASSARD, S.P. HASTINGS: Pattern formation and periodic structures in systems modelled by reaction diffusion equations. Bull. Am. Math. Soc. 84. 6, p. 1296–1327 (1978).CrossRefzbMATHMathSciNetGoogle Scholar
  10. (10).
    J. J. HOPFIELD: Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sc. U.S.A. (79), p. 2554–2558 (1982).CrossRefMathSciNetGoogle Scholar
  11. (11).
    S. KAUFFMAN: Behaviour of randomly constructed genetic nets. in Towards a theoretical biology, Vol3, Edinburgh University Press, p. 18–46 (1970).Google Scholar
  12. (12).
    S. KIRKPATRICK: Models of disordered materials. In Ill condensed matter, Les Houches, North Holland (1979).Google Scholar
  13. (13).
    C.A. MEAD, M.A. CONWAY: Introduction to V.L.S.I systems Addison Wesley, (1980).Google Scholar
  14. (14).
    P. PERETTO: Collective properties of neural networks; A statistical physics approach. (to appear in Biological Cybernetics.)Google Scholar
  15. (15).
    P. QUINTON: The systematic design of systolic arrays. (to appear).Google Scholar
  16. (16).
    F. ROBERT: Discrete iterations Springer Verlag (to appear).Google Scholar
  17. (17).
    Y. ROBERT: Thesis, Grenoble (to appear).Google Scholar
  18. (18).
    R. SHINGAI: Maximum period of 2-dimensional uniform neural networks Inf and Control (11), 324–341, (1979).Google Scholar
  19. (19).
    M. TCHUENTE: Contribution à l’étude des méthodes de calcul pour des systèmes de type coopératif, Thesis, Grenoble (1982). (see also the paper by M. Tchuente in the present volume.)Google Scholar
  20. (20).
    R. THOMAS: Kinetic Logic, Lecture Notes in biomathematics, Vol 29, Springer Verlag (1979).Google Scholar
  21. (21).
    J. VON NEUMANN: Theory of self reproducing automata., A.W. Burks Editor; University of Illinois Press (1966).Google Scholar
  22. (22).
    S. WOLFRAM (Editor): Cellular automata. Los Alamos Science (1984).zbMATHGoogle Scholar
  23. (23).
    S. WOLFRAM: Statistical mechanics of cellular automata. Rev. Mod. phys. 55, n° 3, 601–642 (1983).CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • François Robert
    • 1
  1. 1.IMAG/INPGUniversité de GrenobleSt Martin d’HèresFrance

Personalised recommendations