Chaotic Attraction - Part 4
Boundary conditions recursion scaling and convergence
In contrast
to the Newtonian position in which a given color is reduced to a precise wave length,
colors that arise in the context of perception are more difficult to contain in any such
precise fashion. The horizonal boundaries of perceptual color seem more diffuse and spread
out over a broader phenomenological expanse than are the Newtonian representations of a
color's wave length. Nonetheless, despite the incredible complexities and intricacies
surrounding the problems of boundary conditions in relation to color perception, human
beings are capable of inter-subjectively agreeing, on a consistent basis, concerning the
character of the color generated during perceptual engagements of such boundary
conditions.
Mitchell
Feigenbaum, a physicist, considered the foregoing case involving color perception to be a
crucial example of how order and universality could emerge from apparent chaos and
turbulence. Moreover, this example would serve as a concrete source of inspiration for his
subsequent, more abstract, probing of the relationship between order and chaos in the
context of changing boundary conditions.
Feigenbaum's
starting point for his more abstract studies was the equation for a parabola: namely, y =
r(x-x2). In this equation the variable 'r' determines the steepness of the parabola's
arch.
However, he
wanted to employ the parabolic function in a recursive fashion. That
is, he wanted to take the results from having run one set of values for the variables
through the function and feed those results back into the function. This process of
feed-back would, then, be repeated again and again.
He often
found that recursion yielded a stable state in which x and y would be equal or in
equilibrium. As a result, the character of the graph would not change with further rounds
of feed-back.
On other
occasions, however, the recursion process did not lead to one final, steady, equilibrium
state. On these occasions, Feigenbaum observed that the system fluctuated between several
values.
In general,
Feigenbaum found the behavior of the system was extremely sensitive to the value of 'r'
which determined the steepness of the arch of the parabola. If the value of ‘r’
produced an arch with too little steepness, the end result would be the extinction of
whatever system was being represented. This extinction occurred because the recursion
process would eventually require the equation to produce a value of 0 for y.
On the other
hand, if Feigenbaum permitted the value of 'r' to increase past some minimal, critical
level, the recursion process led to the production of a steady state equilibrium shaped by
a point attractor. Finally, if the value of 'r' were increased further, Feigenbaum
encountered a system which first underwent bifurcation and period doublings. If the value
of 'r' was increased still further, he ended up with a totally chaotic system that was not
attracted toward any points of equilibrium.
In the
hermeneutical context, the hermeneutical operator plays the role which 'r' plays in the
parabola equation cited above. Under different experiential circumstances, the phase state
of the hermeneutical operator, as it is manifested through the focal/horizon dialectic,
constitutes the critical variable. This variable determines whether a given perspective,
orientation or understanding will: (a) become extinct (i.e., produce static or zero values
when plugged into the hermeneutical field equations in recursive fashion); (b) lead to
some sort of equilibrium state defined by a fixed-point or limit cycle attractor; or, (c)
generate a chaotic process which will never settle down to any set of self-same values.
Previously,
Lorenz had discovered that other kinds of possibilities could arise in systems which were
somewhat more complicated than the ones generated from recursion of Feigenbaum's starting
equation. For example, in some cases, one could encounter systems which harbored more than
one stable state.
Under some
circumstances, one of the possible stable states establishes itself and persists for a
long time. However, under other circumstances, an entirely different sort of equilibrium
might establish itself. This sort of system is known as an intransitive system.
In an
intransitive system, one state of equilibrium or another would establish itself, but the
system did not oscillate between the two. Moreover, on any given occasion, which of the
possible states of equilibrium will establish itself depends on the sorts of external
vectoring or shaping factors impinging on the system. Consequently, an
intransitive system only can change the character of the kind of equilibrium it manifests
if forces external to the internal dynamics of the system are capable of bringing about or
inducing such a transition in behavior.
In some
circumstances, a hermeneutical system seems to reflect some of the properties of an
intransitive system. More specifically, in order for a given hermeneutical orientation,
attitude, perspective or understanding, which has achieved a certain degree of
equilibrium, to give rise to a different kind of hermeneutical orientation, attitude,
perspective, etc., one must subject that hermeneutical state to a set of extrinsic forces.
These extrinsic forces must be sufficiently strong, or appropriately structured, to induce
a phase transition toward one of the other hermeneutical states of equilibrium which are
open to a given hermeneutical system.
Furthermore,
in intransitive hermeneutical systems, an individual would be in one, or another, of the
hermeneutical phase states that are possible in such a system. However, one does not
oscillate between them. Which hermeneutical phase state one would be in depends on the
nature of the forces impinging on the individual, together with the manner in which the
individual was hermeneutically oriented toward those forces.
On the other
hand, there might be other hermeneutical contexts in which an individual might oscillate
between several phase states or orientations. For example, if the individual were
vacillating between several possibilities, such as in cases of approach/approach or
avoidance/approach, or something similar, then, the individual's hermeneutical stance
might not display intransitive characteristics.
In addition
to intransitive systems, Lorenz also spoke of systems displaying the property of
almost-intransitivity. This kind of system would display one sort of average
behavior for an extended period. Thus, the behavior would never go beyond a certain
envelope of values. It would fluctuate within this set of values, never settling down to
any specific value.
Inexplicably,
at a certain point, the system suddenly would begin to manifest a different kind of
average behavior as a new envelope of boundary limit values was established due to a new
set of fluctuations within the system. Such systems were highly unpredictable.
In 1975,
Feigenbaum began to explore the problems surrounding the transition from periodicy to
chaos in systems involving a quadratic map. The transition from periodicy to chaos in such
systems seemed to share certain features in common with the way in which laminar flow gave
way to turbulence in fluid systems. More specifically, as one encountered the boundary
region of transition marking the change from periodicy to chaos, one encountered the
tell-tale sign of bifurcation, with its cascade of cycle splitting and doublings.
By
calculating the values of the parameters which led to the bifurcations, Feigenbaum
discovered that the cycle splitting was not just taking place at a faster and faster rate,
they were doing so in a constant manner. In short, the parameters were converging
geometrically. This suggested that some aspect of the system was repeating itself across
scales.
In short,
his findings indicated scaling was somehow involved in the process of generating the
pattern of bifurcations being observed in the context of quadratic mapping. The presence
of the property of scaling meant that in the midst of all manner of transitions and
changes, something was being preserved from one level of scale to the next.
Feigenbaum
calculated the ratio of convergence for the recursion of quadratic mapping. He obtained a
value of 4.669.
He, then,,
began to look at other functions as grist for the recursion mill. One function that he
explored concerned the sine of a number, for which he used the equation: xt+1 = r (sin
)(pi)xt
Once again,
Feigenbaum found that the numbers being generated by the equation during the recursion
process were converging in a geometric manner. In fact, the convergence ratio was
precisely the same as he had discovered in relation to the quadratic map - namely:4.669.
With each
new function he tried, he observed the same property of geometric convergence emerging
when the function was subjected to the recursion process. Even more amazingly, with each
new function he tried, he found the same convergence ratio of 4.669 waiting for him at the
end of the process. Apparently, the order underlying the recursion phenomenon was somehow
independent of the equations being used as a source for number generation.
Prior to
Feigenbaum's discovery, the methodological techniques scientists had used to try to get a
handle on the global or long-term behavior of a physical system depended on a knowledge
and understanding of the mathematical functions being used to represent or model a given
system. However, the rule rather than the exception proved to be that, in the case of
nonlinear systems, use of the aforementioned functions did not help one achieve the sort
of global understanding which had been sought.
In the light
of Feigenbaum's work, there seemed to be universals at work which did not depend on the
specific character of such functions since, irrespective of the function selected, the
recursion process led to precisely the same convergence ratio. Whether one used a
trigonometric function or a quadratic function or some other kind of function, the means
seemed to be irrelevant to the end result.
On the basis
of his discovering of the scaling universal that was at the heart of the recursion
process, Feigenbaum began to look for a different approach to solving nonlinear problems.
He was looking for an approach rooted in the new universal convergence ratio rather than
any particular kind of mathematical function. One of the first places his explorations
took him was to the study of attractors.
The points
of equilibrium to which his mappings gave expression were attractors. These
attractors shaped or constrained the fluctuations of a system regardless of the starting
point from which the system began. However, when a system undergoes bifurcation as a
result of continued recursion, the attractor splits in two.
As the
system is subjected to further rounds of recursion, the split attractor points would begin
to grow more distant from one another until a further bifurcation would occur. This
resulted in each of the attractor points splitting yet again and at precisely the same
time.
Feigenbaum's
universal convergence value enabled him to predict when the various bifurcations would
occur. In addition, he found that he also was able to predict the precise value of the
point where these values would occur on the attractor as it became increasingly more
structurally complex with each new round of recursion.
The
phenomenon Feigenbaum had stumbled upon showed strong indications of being
self-referential and recursive, as well as exhibiting multiple-scaling properties.
However, largely because of its scaling features, Feigenbaum decided to apply
re-normalization group theory which had provided physicists with a means of canceling out
the embarrassing infinities that kept surfacing in quantum mechanics. Apparently, he felt
that if re-normalization group theory could use scaling techniques to resolve one set of
problems in physics, perhaps, one could make use of these same scaling techniques to
provide insight into the universal principles underlying the multiple-scaling
characteristics of the nonlinear systems he had been exploring.
Even though
Feigenbaum was dealing only with simple mathematical functions, he believed his numerical
recursion experiments revealed a law of nature inherent in any system, mathematical or
physical, which was at the boundary between turbulence and order. In other
words, when conditions in a system began to generate the bifurcations and period doublings
characteristic of turbulence, a spectrum of frequencies would emerge.
The etiology
of this spectrum had always baffled investigators. Feigenbaum's universal convergence
ratio, however, seemed to provide a window through which to observe the coming into being
of the spectrum of frequencies which heralded the transition from orderly to nonlinear
behavior in a given system.
In essence,
Feigenbaum was drawing attention to the existence of structural features in nonlinear
systems which were preserved across all levels of scale in such systems. He was talking
about a symmetry property in the midst of seeming chaos. All one had to do to observe the
presence of this property was look at the system in the right methodological way.
Feigenbaum's
conclusion can be translated into the perspective of the present article. An order-field
gives expression to the constraints, degrees of freedom and transitions in the spectrum of
ratios of constraints and degrees of freedom that are manifested on all levels of scale.
The mutual
penetration of chaos in symmetry and symmetry in chaos is a function of the dialectic of
dimensionality. What we experience as material, physical or mental latticeworks on the
macro level are really scale independent, fractal expressions of the dialectic of the
constraints and degrees of freedom of the basic "stuff" of the everyday world of
experience: namely, dimensionality.
The
universal convergence ratio observed by Feigenbaum is given expression, to some extent,
through the manner in which the hermeneutical process proceeds toward a merging of
horizons with some aspect of ontology and/or the phenomenology of the experiential field.
In other words, the hermeneutics of experience involves applying an algorithm (i.e., the
hermeneutical operator) in a recursive fashion in order to produce hermeneutical maps,
just as Feigenbaum needed to generate a recursive function to produce his quadratic maps.
Because the
hermeneutical mapping algorithm is a far more complex function (i.e., a tensor-matrix)
than any of the mathematical functions studied by Feigenbaum, it will not yield the
numerical convergence ratio which emerged again and again in the different recursive,
mathematical mappings undertaken by Feigenbaum. The numerical convergence ratio is a
reflection of the invariant principles that are inherent in the structural character of
mathematics as a methodology.
On the other
hand, the hermeneutical mapping algorithm does give expression to methodological
structures that all exhibit invariant principles inherent in the character of the
hermeneutical operator as manifested in the form of a spectrum of ratios of constraints
and degrees of freedom. These principles are independent of any particular application of
the hermeneutical operator, just as Feigenbaum's numerical convergence ratio is
independent of the particular identity of the mathematical function being recursively
mapped.
Moreover,
just as the numerical convergence ratio is the structural signature of certain principles
that are operative at the heart of mathematics, so too, the hermeneutical operator is the
structural signature of the principles that are operative at the heart of understanding.
In fact, Feigenbaum's numerical convergence ratio is but a specialized exemplar of the
hermeneutical operator in action since the former is the product of a particular set of
hermeneutical mapping algorithms which have been recursively applied to a particular issue
- namely, the relation of turbulence to laminar flow.
Contiguity in complex boundaries
Michael
Barnsley, an English mathematician, was interested in trying to discover the reasons
underlying the period doublings of Feigenbaum's convergence sequences. He believed such
sequences must be linked to a fractal phenomenon in some way which had not, yet, been
fully grasped by investigators.
Moreover, he
felt exploration could best be conducted through the complex plane encompassing both the
real numbers as well as the imaginary numbers. In fact, real numbers are a
special kind of complex number in which the value of the imaginary component is zero and,
therefore, can be ignored.
When
Barnsley introduced Feigenbaum's convergence sequences into the complex plane, a
intriguing set of structures were generated. As it turned out, however, the structures
Barnsley had uncovered already had been discovered some 50 years prior to his work. They
were known as Julia sets.
According to
Barnsley, if one slices a round cake into a number of pieces, all of the pieces converge
at a common point in the center. Furthermore, provided one idealizes conditions somewhat
(such as assuming that the side faces of each slice are perfectly smooth), the boundary
between any two of these slices is relatively uncomplicated.
However,
Barnsley continues, if one were to perform a comparable sort of operation in the complex
plane, one would get a much different result. In fact, the boundary between any two given
slices of a complex plane cake would become incredibly convoluted. Every point on a
boundary separating slices would be in contact with other slices of the cake as well.
Apparently,
a solid boundary never forms in the realm of complex numbers as occurs in the
three-dimensional world. When one inspects the complex boundary in closer detail, it seems
to dissolve into complex remnants of the other regions which have been sliced up. Thus,
this mysterious property of, for lack of a better word, 'convolution' or 'permeation' or
'convoluted permeation', would hold across all levels of scale, thereby, displaying
fractal characteristics.
One might
well suppose that the dialectic between focus and horizon, is capable of giving expression
to boundaries with this same sort of convoluted complexity as the complex plane. If this
is the case, then, any given point of a hermeneutical horizon is capable of bordering on
all the aspects of a given focus and vice versa. Which aspects of horizon and focus will
be given expression will depend on how the hermeneutical operator links horizon and focus
together through the exchange of hermeneutical phase quanta.
Furthermore,
one might suppose that hermeneutical systems manifest periodic or aperiodic bifurcations
as such a system is pushed into turbulence. The turbulence is the result of the problems,
questions, enigmas, and so on which arise during the course of the individual's attempt to
come to terms with the hermeneutics of various aspects of experience.
The
phenomenon of aperiodic bifurcations also emerges on a social/historical level of scale.
For example, Thomas Kuhn's account of the breakdown of 'normal' science and the emergence
of revolutionary science seems to reflect a great deal of the flavor of the multiplicity
of interpretive frameworks which arise during periods of crisis.
Eventually,
however, one of the candidates gets selected to assume the throne in order to direct the
reign of a new expression of normal science. Therefore, the transition from
normal science to revolutionary science appears to exhibit many of the characteristics of
a system undergoing turbulence, bifurcations, chaos and, finally, a new kind of order
which emerges out of the chaos.
Thus, the
period doublings of physical systems seem to have their hermeneutical counterparts in the
way possible solutions (whether from an individual or from a scientific community) tend to
multiply as the focal/horizonal dialectic seeks to discover a viable solution to some
outstanding anomaly, problem, question, issue or challenge. A solution or resolution
emerges if one can establish a function of the appropriate sort of convergent character.
In the
context of hermeneutical field theory, this appropriate sort of function would assume the
form of a tensor-matrix generated by the recursive activity of the hermeneutical operator
as it produces structures capable of merging horizons, at least on some levels of scale,
with various aspects of ontology or phenomenology. Under such circumstances, order emerges
out of turbulence.
Chaos: a science of being and becoming
Chaotic
dynamics is considered by some physicists to be a science of process rather than a science
of state, or a science of becoming rather than a science of being. However,
there really is no reason why one couldn't consider it to be a science of both process and
state, or becoming and being.
Chaos gives
expression to the properties of dynamics and dialectics through its aspects of process and
becoming. Simultaneously chaos gives expression to the properties of structure and
latticework (as an envelope of a set of constraints and degrees of freedom) through its
aspects of state and being.
Moreover,
whether being expressed as process or state, becoming or being, all of these are
manifestations of the underlying order-field. This order-field establishes and regulates
the dialectic of dimensions that generate the dynamics of a given latticework or set of
latticeworks.
Therefore,
the order-field determines the structural character of the spectrum of constraints and
degrees of freedom that are capable of giving expression to chaotic dynamics under the
right set of circumstances. This is the case in both physical as well as hermeneutical
systems.
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