At the end of the sixties, our program for dynamics lost momentum, and some of us turned to applications for inspiration. My own attempts at modeling in the biological sciences led me to complex dynamics, a blend of our own style of dynamical systems theory with system dynamics. A complex dynamical system is a directed graph, with a dynamical scheme (dynamical system with parameters) at each node, and a coupling function on each edge, expressing the control parameters of one scheme as a function of the states of another.1


Coupling Function Standard Cell Dynamical System Theory Complex Dynamical System Dynamical Scheme 
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. Ralph H. Abraham, Complex dynamical systems, Mathematical Modelling in Science and Technology, eds. X.J.R. Avula, R.E. Kalman, A.I. Leapis, and E.Y. Rodin, Pergamon Elmsford, NY, pp. 82–86. 1984Google Scholar
  2. Ralph H. Abraham, Cellular dynamical systems, Mathematics and Computers in Biomedical Applications, Proc. IMACS World Congress, Oslo, 1985, eds. J. Eisen-feld and C. DeLisi, North-Holland, Amsterdam, pp. 7–8. 1986Google Scholar
  3. Ralph H. Abraham, Cuspoidal nets, Toward a Just Society for Future Generations, eds. B.A. and B.H. Banothy, Int’l Society for the Systems Sciences, pp. 66–683. 1990aGoogle Scholar
  4. Ralph H. Abraham, Visualization techniques for cellular dynamata, Introduction of Nonlinear Physics, ed. Lui Lam, Springer-Verlag, Berlin. 1990bGoogle Scholar
  5. Ralph H. Abraham, Gottfried Mayer-Kress, Alexander Keith, and Matthew Koebbe, Double cusp models, public opinion, and international security, Int. J. Bifurcations Chaos 1(2), 417–430. 1991aMathSciNetMATHCrossRefGoogle Scholar
  6. Ralph H. Abraham, John B. Corliss, and John E. Dorband, Order and chaos in the toral logistic lattice, Int. J. Bifurcations Chaos 1(1), 227–234. 1991bMathSciNetMATHCrossRefGoogle Scholar
  7. Ralph H. Abraham, Cellular dynamata and morphogenesis: a tutorial, preprint. 1991cGoogle Scholar
  8. Martin Beckmann and Tonu Puu, Spatial Economics: Density, Potential, and Flow, North-Holland Amsterdam. 1985Google Scholar
  9. Martin Beckmann and Tonu Puu, Spatial Structures, Springer-Verlag, Berlin. 1990Google Scholar
  10. James P. Crutchfield and Kunihiko Kaneko, Phenomenology of spatio-temporal chaos, Directions in Chaos, ed. Hao Bai-Lin, World Scientific, Singapore. 1987Google Scholar
  11. Walter Freeman, The physiology of perception, Sci. Am. 264 (2), 78–85. 1991MathSciNetCrossRefGoogle Scholar
  12. Wulf Gaertner and Jochen Jungeilges, A model of interdependent consumer behavior: nonlinearities in R2, preprint. 1991Google Scholar
  13. Morris W. Hirsch, Convergent activation dynamics for continuous time networks, Neural Networks 2, 331–349. 1989CrossRefGoogle Scholar
  14. Kunihiko Kaneko, Simulating Science with Coupled Map Lattices, Formation, Dynamics, and Statistics of Patterns, ed. K. Kawasaki, A. Onuki, and M. Suzuki, World Scientific, Singapore. 1990Google Scholar
  15. Nancy Kopell and G.B. Ermentrout, Coupled oscillators and mammalian small intestines, Oscillations in Mathematical Biology, ed. J.P.E. Hodgson, Springer-Verlag, Berlin, pp. 24–36. 1983Google Scholar
  16. Steve Smale, A mathematical model of two cells via Turing’s equation, The Hopf Bifurcation and its Applications, eds. J.E. Marsden and M. McCracken, Springer-Verlag, New York. 1976Google Scholar
  17. E. Christopher Zeeman, Duffing’s equation in brain modelling, Catastrophe Theory, ed. E. Christopher Zeeman, Addison-Wesley, Reading, MA, pp. 293–300. 1977Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • R. Abraham

There are no affiliations available

Personalised recommendations