Page 275 - Mechatronics for Safety, Security and Dependability in a New Era

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be well-defined, and thus it inevitably moves to its opposite, the other whole. This cyclic movement
never stops or, rather required not to stop on this ceaseless flow of dialectics, in order that the
reciprocal wholes both should be defined.

Figure 1: Model for Dynamic Unification of the Acting Potential and Event

The dynamic interaction goes way beyond dynamics of events, physical or otherwise. It is the more
fundamental movement between the two wholes that molds both events and potentials with its
dynamic framework. It is not just logically anticipated, but governing principle of reality, more akin to
Heraclites' proposition in antiquity "all is in a state of flux" (Russell 1945). Tt also gives the
substantial ground why the opposite things interact at first place, A and ~A, opposing alternatives
which press on decision makers under impending pressure both in the domains of potentials and of
events. The potential mode of existence is particularly relevant to decision making, where the opposite
potencies are both rushing toward realization as event, but only one of which will be realized
exclusively.

One of the simplest equations among possible others which entertains the Mobius model is the
Verhulst equation, x,,+i = b x n • ~x n (Verhulst 1845, Feigenbaum 1978) . It is not only relevant to the
original application for the growth of populations, but for the rather far-reaching extension of
application, that is describing the deterministic interaction of opposites in the process of decision
making. The Verhulst equation expresses iterative interplay of reciprocalities of two kinds, additive
and multiplicative (x,, + ~x n = 1 and x,, ' ~x n = ^ n +\ (= x n+iIb ), respectively) at the right hand side of
the equation. Both of them equally satisfy the defining relation of reciprocality among the quantities of
two or more variables in a way that when one quantity increases, the other decreases or the other way
around, though their quite distinct ways of increasing or decreasing.
The Verhulst equation embodies a representation of the iterative fundamental movement between two
distinctive wholes, the domains of events and of potentials by capturing the interplay of both types of
reciprocals, x • ~x =1 and x + ~x = 1. Such iterative interplay between both types normally leads to a
complex behavior as depicted at Figure2. The equation consists of a series of steps of transformations,
where the fundamental movement between two wholes governs along the Mobius surface (Eqn.l); An
event x,, at n"' generation occurs in the domain of events, and determines it's unrealized opposite \-x n
(= ~x n). The opposite then moves to the domain of intension or potential, where both x n and ~x n reside
as opposite potentials in the form of x n'~x n. The potential then produces an event of n+\ th generation
by the dynamic law of Verhulst, x n+i=bx,,'~x n. (Note: The additive reciprocality, x n + ~x n = 1 expresses the
sum of opposites is the whole or "the whole is the sum of parts". It is the distinctive characteristic of extension. It
does not hold for the intensional whole which completely lacks extension. The multiplicative reciprocality, x r • ~x rl
— 1 is rather "the intensional whole is the product of parts", for the intensional parts of attributes are all enfolded in
one entangled state of the intensional whole. This entanglement establishes a product as the natural operator for the
domain of the intensional whole, where an essential non-linearity reigns.)