The process equation consists of a sequence of actions At. Each action At occurs at specific moment in time t. The concept of action as a change of energy in time applies to all levels of organization. Action is defined
in physics as the integral of energy change in time. Action, rather than energy, is the simplest entity to be considered,
because the Planck constant, the simplest possible change, has the dimensions of action.
Every action At+1 at time t
is a function of the previous action At,
portraying the continuity of the process. This continuity is composed of discrete steps. Actions are discrete at
all levels of organization: Planck's quantum of action, mechanical movement, cardiac contractions, behavioral action
patterns. Correspondingly, this is a difference equation rather than a differential equation.
The sequence of actions is modeled by the time series of actions At. Actions are ordered in time. At+1 is a function of At
but At is not a function of
At+1.
Each action At+1 continues the
previous action At but modified
by change (At. Change results from the interaction of the process under study with other processes; thus (At often is a function of the preceding action
At, even if the external process
is, or appears to be, random. Repetitive interactions thus represent feedback.
Interactions can be positive (augmenting, synergistic) or negative (decreasing, antagonistic). Given sufficient
time and/or extension, the set of all interactions includes both, certainly so in the case of a random milieu.
The feedback is bipolar, not only
positive or only negative; further, it takes a wide variety of values. Trigonometric functions such as (At = g sin At model processes that include
both positive and negative interactions; this feedback is bipolar and diverse, spanning a continuous range of values
from one pole to its opposite. This harmonic feedback may thus model the notion of complementary opposites.
The equation is computed in two different forms: (1) dynamic: the gain g is kept
constant, and 200 or more initial iterations are discarded (figures in next page); (2) kinetic: the gain g is a
function of time, such as g = k * t, for some small k (first page figures). There are significant quantitative
differences between the dynamic and kinetic equation regarding critical values of g. In either case, as the gain
g increases, the equation generates a series that progresses from simple to complex patterns in a manner resembling
cosmological evolution and embryological development:
Main patterns generated by the process equation. The graphs present
the first 500 data points calculated at constant g, from a set of initial values ranging from 3 to -3 . Top left:
Divergence to multiple steady states. Right: bifurcation to period 2. Middle left: Chaos. Right: Bios. Bottom left:
infinitation. Right: Return to biotic phase.
(1) Convergence: For g < 2, the equation converges to an even multiple of pi,
a single initial, asymmetric (non-zero), steady state.
(2) Bifurcations: At g > 2, it bifurcates into complementary opposites that
diverge; when the lower path approximates 1.6181.. (Fibonacci's ratio describing spiral order), the upper path
approximates 4.6692..., suggesting that these two constants are related to each other, and represent bifurcations
of p. When g = pi, there is a unifurcation. This is followed by a cascade of bifurcations generating period 4,
8 ...
(3) Chaos
emerges when g = 3.56... (Feigenbaum's point in the logistic equation). Process chaos is interrupted by periodicities,
and shows "walls" not observed in other chaotic processes; it is highly anti-correlated (Pearson's correlation
-0.99)
(4) Bios, an apparently erratic pattern of much larger magnitude and highly self-correlated (Pearson's
coefficient +0.99), emerges at g = 4.604. This biotic pattern closely resembles the pattern of heart rate variation. At g = 4.6692..
(Feigenbaum's constant), bios changes, departing from the pattern observed in heartbeat series; this change is
dramatically illustrated by rational plots. Bios is interrupted by bioperiodicities extremely sensitive to initial
conditions.
(5) Infinitations: escapes towards positive or negative infinity interrupt bios,
whenever g = 2n pi, and at few other critical values such as g = 0.5 * pi2 and 23).
The next two figures illustrate how these various patterns evolve with increasing
g. The figures illustrate the time series generated by the kinetic process equation (g increasing with time). The
top figure illustrates the early patterns at a low g, while the bottom figure illustrates those patterns with much
higher amplitude generated by higher gains (g).
The time series generated by the kinetic equation, illustrated above (increasing
g), is very similar to the time series generated by the bifurcation diagram of the dynamic process equation (constant
g) This equation diagram illustrates the greater creativeness of bipolar feedback, both synergic and antagonistic,
in comparison to simple positive or negative feedback.
Particularly noteworthy is the generation of numerical archetypes, 2, pi, phi, etc. These numerical archetypes
embody various aspects of opposition: a gain of 2 produces bifurcation, and pi is the ratio of circular opposition
to linear order.
The process equation formulates the fundamental principle of process theory: interpreting At as action, iteration
and addition embody the unidirectional progress of time, and the sine function, sometimes positive and sometimes
negative, represents complementary opposites.
Action and information are the complementary opposites that co-create pattern in the process equation. The temporal
order of action is a universal asymmetry, more fundamental than mechanical reversibility. It is complementary to
the symmetry of information, the simplest and fundamental form of which is the complementarity of opposites. Action
is linear and information is circular. The relation between circle and line is p, historically the first, and perhaps
the most fundamental numerical constant. Significantly, p appears as the fixed point in the process equation as
well as in many other roles.
The process equation thus shows how unidirectional action and complementary
opposition are sufficient to create novelty, diversity, and complexity
(co-creation). Co-creation by complementary opposites may thus account for creative evolution without resorting
to random accident or supernatural intervention.
The process equation models the most general type of process. A number of
process equation variants explore various aspects of this general model,
and the diversity, complexity and creativity that stems from bipolar feedback, including the generation of organic patterns.
