Process equation

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A(t+1) = A(t) + g \sin(A(t)) \,.

Although variations of this equation have been studied before, Louis Kauffman and Hector Sabelli[1], [2] thought that it modeled well processes in nature, and have named it accordingly.

When g is kept constant, then, depending on its value, this recursion generates either a steady state, periodicity, chaos, bios or infinitation (output increases in size toward infinity). When g = kt, where k is a small constant, this recursions generates all above patterns on different values of g. Without a conserved term, A(t), this recursion cannot produce bios. This recurrence relation is related to the circle map.

Steady state, bifurcations, chaotic phase, and bios in development of recursive equation. Note that y-axis does not show expanding series, so that bifurcations and chaos could be seen.
Steady state, bifurcations, chaotic phase, and bios in development of recursive equation. Note that y-axis does not show expanding series, so that bifurcations and chaos could be seen.
Transition from chaos to bios (detail from the image on the left with y-axis showing the full range of the series and x-axis focusing on the transition/expansion of the series).
Transition from chaos to bios (detail from the image on the left with y-axis showing the full range of the series and x-axis focusing on the transition/expansion of the series).


[edit] References

  1. ^  Kauffman, L. and Sabelli, H. (1998) The Process equation. Cybernetics and Systems 29: 345-362
  2. ^  Sabelli, H. and Kauffman, L. (1999) The Process equation: Formulating and Testing the Process Theory. Cybernetics and Systems 30: 261-294.
  3. ^  Kauffman, L. and Sabelli, H. (2003) Mathematical Bios. Kybernetes 31: 1418-1428.

[edit] External links