Discrete Dynamics in Nature and Society
Volume 1 (1997), Issue 1, Pages 1-8

Discrete dynamics of complex systems

Hermann Haken

Institute for Theoretical Physics and Synergetics, University of Stuttgart, Pfaffenwaldring 57/4, Stuttgart D-70550, Germany

Received 9 October 1996

Copyright © 1997 Hermann Haken. 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.


This article extends the slaving principle of synergetics to processes with discrete time steps. Starting point is a set of nonlinear difference equations which contain multiplicative noise and which refer to multidimensional state vectors. The system depends on a control parameter. When its value is changed beyond a critical value, an instability of the solution occurs. The stability analysis allows us to divide the system into stable and unstable modes. The original equations can be transformed to a set of difference equations for the unstable and stable modes. The extension of the slaving principle to the time-discrete case then states that all the stable modes can be explicitly expressed by the unstable modes or so-called order-parameters.