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Iterations

  • Wolfgang Kinzel
  • Georg Reents

Abstract

A function consists of a set of instructions that convert given input values to output values. Now if the output itself is part of the domain of the function considered, it can become an input in turn, and the function returns new output values. This process can then be repeated indefinitely. While there are often no analytic methods to calculate the structural properties of such iterations of the form x t +1 = f(x t ), they are easily generated on the computer. Obviously, one just has to apply the same function f(x) to the result repeatedly. We want to demonstrate this with a few examples.

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Literature

  1. Crandall R.E. (1991) Mathematica for the Sciences. Addison-Wesley, Redwood City, CAGoogle Scholar
  2. Bai-Lin H. (1984) Chaos. World Scientific, SingaporezbMATHGoogle Scholar
  3. Jodl H.-J., Korsch H.J. (1994) Chaos: A Program Collection for the PC. Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  4. Ott E. (1993) Chaos in Dynamical Systems. Cambridge University Press, Cambridge, New YorkGoogle Scholar
  5. Peitgen H. O., Saupe D. (1988) The Science of Fractal Images. Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  6. Schroeder M. (1991) Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W. H. Freeman, New YorkGoogle Scholar
  7. Schuster H.G. (1995) Deterministic Chaos: An Introduction. VCH, Weinheim, New YorkzbMATHGoogle Scholar
  8. Griffiths R.B. (1990) Frenkel-Kontorova Models of Commensurate-Incommensurate Phase Transitions. In: van Beijeren H. (Ed.) Fundamental Problems in Statistical Mechanics VII. Elsevier, Amsterdam, New York, 69–110Google Scholar
  9. Greene J.M. (1979) A Method for Determining a Stochastic Transition. J. Math. Phys. 20: 1183ADSCrossRefGoogle Scholar
  10. Peitgen H.O., Saupe D. (1988) The Science of Fractal Images. Springer, Berlin, Heidelberg, New YorkzbMATHGoogle Scholar
  11. Schroeder M. (1991) Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman, New YorkzbMATHGoogle Scholar
  12. Wagon S. (1991) Mathematica in Action. W.H. Freeman, New YorkzbMATHGoogle Scholar
  13. Hertz J., Krogh A., Palmer R.G. (1991) Introduction to the Theory of Neural Computation. Addison-Wesley, Reading, MAGoogle Scholar
  14. Müller B., Reinhardt J., Strickland M.T. (1995) Neural Networks: An Introduction. Springer, Berlin, Heidelberg, New YorkCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Wolfgang Kinzel
    • 1
  • Georg Reents
    • 1
  1. 1.Institut für Theoretische PhysikUniversität WürzburgWürzburgDeutschland

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