Chaotic Attraction - Part 3
Attractors in the complex plane
A major
difference between, on the one hand, geometric structures like the Sierpinski sponge or
the Koch curve and, on the other hand, natural structures like the coastline of Britain,
is that the former are highly regular or predictable, whereas the latter tend to be more
irregular and less predictable. The primary reason for the regularity of constructed
structures such as the Koch curve or the Sierpinski sponge is that they are invariant
under the sorts of simple scaling, linear transformations applied to them during the
generative process which leads from simple geometric figures to their fractal
counterparts.
However,
naturally occurring figures, such as a coastline, may not remain invariant - at least in
any linear fashion - under the scaling transformations that are applied to them
. Even if such figures do remain invariant, then, one has the problem of determining
whether the invariance is an artifact of a methodology’s manner of engaging some
facet of ontology, or a reflection of some aspect of ontology which is being engaged
through a given methodology.
Around the
mid-1970s, Mandelbrot began to investigate a more complex form of fractal-generating
processes involving invariant properties under conditions of transformations. Eventually,
toward the close of the 1970s, Mandelbrot settled on the complex function, x2 +
c, to serve as a main vehicle for his computer researches. In the foregoing function, both
the constant, c, as well as the variable, x, represent complex numbers.
A feedback
loop involves a process of change for some given structure, q, in which the changing
character of that structure in the present is dependent on the way that structure
manifested itself just prior to the present instant. The nature of this dependency is
usually in the form of a mathematical function, f(x). This function establishes how one
will take some seed value, x0, and, then,, proceed to generate successive
values: x0 + 1, x0 + 2, x0 + 3 and so on.
Feedback
loops which map an initial seed value into a series of successive values are usually
referred to as dynamical systems. The set of successive values into which the
seed value is mapped is known as a path or orbit.
If this
orbit or path is ordered, then, the associated dynamical system is said to be ordered. If,
on the other hand, the mapping path is not ordered, then, the dynamical system is called
chaotic.
Using the
formula f(x) = x2 + c, and priming this formula with a complex number seed
value, Mandelbrot iterated the function by means of the rule: xn + 1 = f(xn).
He found that the character of the constant, c, had considerable capacity to shape the
kind of results one would obtain from iterating the initial function.
If one
considers the case when c is 0, then, depending on the character of the seed value with
which one begins, there will be three possible results. If the seed value is less than one
unit of distance from the origin, the path generated by mapping this seed value into a set
of successive points will approach the origin as a limit, becoming, as a result,
increasingly smaller with each new iteration of the underlying function. Under such
circumstances, the origin, or 0, becomes an attractor for this sort of an iterated
function.
On the other
hand, if the seed value used to prime the function is greater than one unit of distance
from the origin, then, the path generated will become increasingly further removed from
the origin, tending toward infinity. As a result, infinity is said to be the attractor for
such a feedback loop in the sense that successive values generated through the iterated
function will be become increasingly large, as if drawn toward infinity which is beyond
the horizons of the complex plane.
The final
possibility to consider when c = 0, is when the seed value which is fed into the function
is one unit of distance from the origin. Another way of stating this is to say that the
seed value falls somewhere on the unit circle whose center is the origin of the complex
coordinate system. If this is the case, then, the generated path or orbit will never go
beyond the perimeter of the unit circle.
Consequently,
when c = 0 and x0 = 1, then, the unit circle becomes the boundary between
instances in which x0 is less than 1 unit of distance from the origin, and
instances in which x0 is more than one unit of distance from the origin. In
other words, the unit circle becomes the boundary between the attractors governed by 0 and
infinity.
However,
when the value of the constant c becomes non-zero, rather than 0 as in the foregoing,
Mandelbrot discovered that not only could attractors give expression to more than one
point of attraction, but the boundary structure between such points could become quite
complex. This complexity often manifested itself in the form of structures which were
self-similar, but on a reduced scale, to the object undergoing successive transformations.
This result
already had been anticipated by Julia and Fatou many years earlier when they demonstrated
that any portion or segment of a boundary, irrespective of its size, could be used as a
source of information from which the entire curve or structure of which it was part could
be constructed. All one had to do was use the same iterative function that generated the
system originally.
In the
present case, that function is f(x) = x2 + c. The sequence of numbers generated
in this fashion is referred to as a Julia set.
The
Mandelbrot set is a subset of the complex plane. It has proven to be of
considerable interest due to its apparently inherent rootedness in dynamical processes in
general, or vice versa. More specifically, depending on the character of the value c [cf.
f(x) = x2 + c] in relation to the Mandelbrot set, a number of outcomes are
possible when the foregoing complex function is subjected to a process of iterative
feedback.
For example,
when given a particular value of c relative to the Mandelbrot set, a complex dynamical
process can lead to the degeneration of the Julia set into a set that does not border any
interior sector. Given another value of c relative to the Mandelbrot set, a
complex dynamical process can bring about the division of the complex plane into either
one or more interior sectors, together with an exterior region that extends to infinity.
In short, if
one chooses c from within the main body of the Mandelbrot set, or if one selects a value
of c that is drawn from one of the buds that is connected to the main body of the
Mandelbrot set, or if one focuses on a value of c that falls outside the Mandelbrot set
altogether, then, the structural character of the fractal object(s) one generates during
the iterative process will be differentially affected as a result of the selection one
makes for c relative to the Mandelbrot set.
Hermeneutical mapping algorithms
If one wants
to: establish, dialectically engage, preserve, question and/or, eventually, improve upon
any given set of ideas or values, one must generate hermeneutical mapping algorithms. Such
algorithms are capable of arranging or combining the six basic hermeneutical operations
(identifying reference, characterization, reflexive consciousness, interrogative
imperative, inferential mappings, congruency functions), into a methodological latticework
which can be applied to the phenomenology of the experiential field.
The
hermeneutical operator is an analog for Mandelbrot's function: f(x) = x2 + c.
As such, it is capable of generating attractors whose boundary properties will depend on:
(a) the experiential seed values fed into the operator, together with (b) the
hermeneutical orientation and character of the algorithm that has been constructed by the
individual. The latticework generated by applying the hermeneutical mapping algorithm to
the phenomenology of the experiential field is the hermeneutical counterpart to the notion
of a path or orbit in dynamical systems.
Hermeneutical
mapping algorithms also are recursive. In other words, the products which are generated by
applying hermeneutical operations to different point-structures or seed values drawn from
the phenomenology of the experiential field can be fed back into the hermeneutical
algorithm, thereby altering the character of the way the algorithm operates on future
point-structures in the phenomenology of the experiential field.
Hermeneutical
orientation, together with that to which a given orientation is making identifying
reference, constitute the two ends of the mapping process which is being constructed
through, in part, the operational activity of the algorithm. The mapping itself is an
expression of the dialectic between, or among, the phase relationships of the latticeworks
involved in the dialectical engagement process.
In the
hermeneutical algorithm each of the operational components contribute to the overall
structural character of the algorithm by giving expression to envelopes of constraints and
degrees of freedom. These envelopes establish a latticework of phase relationships that
will engage the 'object', event or condition in a way which is characteristic of that
operational component.
Thus, the
character of the interrogative latticework is to induce questions about phase
relationships and structural themes. On the other hand, the character of the inferential
function latticework is to lay down tentative links between, or among, different aspects
of one or more point-structures. Each of the other components of the hermeneutical
operator has its own characteristic properties.
However, one
must not forget that these operational latticework components cannot really be separated
from one another. They are dialectically entangled such that each forms part of the
horizon of the other. Therefore, they modulate, vector and tensor one another on a
constant basis.
In this
sense, all of these operational components constitute complex point-structures in the
larger, whole, integrated latticework of the hermeneutical mapping algorithm. Thus, one
has latticeworks within latticeworks, and, indeed, one could discover new point-structures
and latticeworks as one went either up or down across various levels of scale.
Each
component operation of the algorithm is "potentially" able to spontaneously
engage an 'object' independently of any considerations except those which pertain to
helping the individual to grasp, with some degree of undistorted reflectivity, the
structural character of the 'object' in question. However, as the word
"potentially" suggests, the hermeneutical mapping algorithm is vulnerable to a
wide variety of influences capable of disrupting its capacity to establish a merging of
horizons.
Said in a
slightly different way, the hermeneutical mapping algorithm is extremely sensitive to
initial conditions. Therefore, the hermeneutical algorithm is susceptible to being pushed
into intense turbulence and/or chaotic behavior which may have little, or no, heuristic
value.
The basic
function of the hermeneutical mapping algorithm is to generate phenomenological models
(ideas, theories, etc) capable of reflecting, in analog fashion, the structural character
of various aspects of ontology being engaged on whatever level of scale that algorithm is
employed. The hermeneutical mapping algorithm is a methodological means of working toward
the unraveling of certain ontological structural themes which are given expression through
the phenomenology of the experiential field. A hermeneutical algorithm is successful to
the extent it terminates in a merging of structural horizons between understanding and
that to which the understanding is making identifying reference in the phenomenology of
the experiential field, as well as that which makes possible a phenomenology of such
structural character.
One of the
central tasks of education is to help the individual discover the structural character of
the hermeneutical mapping algorithm. A further central task of education is to help
nourish that algorithm and permit it to flourish, develop and become refined.
Alternatively, one of the biggest problems of educational theory and practice is to try to
determine exactly when the educational process is interfering with, and/or distorting,
and/or placing unnecessary limitations on the development of the hermeneutical algorithm
which is intrinsic to human beings.
Everything
that is done or attempted in education will be substantially affected by what occurs with
respect to the nurturing or inhibiting of the intrinsic hermeneutical mapping algorithm.
If this algorithm is permitted to be developed properly, the prospect of distortion and
error entering into other aspects of the educational process is a far less likely to
occur. Among other things, if the integrity of the hermeneutical mapping algorithm is
intact, one has a self-correcting means of methodologically protecting oneself against
such distortion and error.
The property of self-similarity in relation to fractal structures
According to
Mandelbrot, there is a link between a system being able to manifest infinitely complex
variations on some given shape/theme and the same system being able to exhibit persistent
characteristics of irregularity. This link is rooted in the property of symmetry to which
fractal structures gave expression.
In other
words, fractal figures have the capacity to preserve the quality of self-similarity from
one level of scale to another. This property of symmetry joins together the aspects of
complexity and irregularity in a dialectic of unified structural character.
The fractal
property which preserves the quality of self-similarity across different levels of scale
is a nonlinear form of recursion theory. Figures such as a Koch curve, or Peano curves, or
Sierpinski carpets, and so on, are recursive structures because the self-similarity aspect
is inherent in the repetitive character of the construction process.
The complex
structural character of the surfaces of materials often prevents different materials that
are brought together from being able to make contact at every point of their respective
surfaces. However, the extent to which interacting surfaces will make contact turns out to
be independent of the materials involved in the interaction. In fact, the dialectic which
establishes the structural character of the contact between (or among) two (or more)
surfaces is a function of the fractal properties of the surfaces involved in the
interaction.
The
Humpty-Dumpty effect is a direct reflection of the way in which contact between surfaces
depends on the fractal properties of the surfaces involved. This effect
refers to the fact that once an object (such as a bowl, cup or glass) is broken, then,
even if one can reconstruct the object so that it appears to fit together on some gross
level of scale, nonetheless, on less gross levels of scale, there will be incongruencies.
This is
because the surfaces have been irreparably altered in various ways by the process of
breaking. As a result, stress bumps are formed where portions of the surface have been
crushed, squeezed and subjected to shearing forces.
Hermeneutics and the Humpty-Dumpty Effect
One might
suppose there will be something like a Humpty-Dumpty Effect in the context of
hermeneutics. In other words, as a result of the impact of ontology on methodology, as
well as a result of the impact of methodology on ontology, fracture zones or zones of
stress will emerge in the realm of understanding.
More
specifically, where the manifold of methodology comes into contact with the manifold of
'reality', the stresses, forces, frictions, limitations, and so on, occurring as a result
of the dialectic of these manifolds, will prevent perfect congruencies from being
established. Consequently, on one or more levels of scale, there will be lacunae and/or
stress bumps which act as obstacles to a total merging of horizons.
In fact, the
limitations which, inevitably, are inherent in any given methodology, have a distorting,
squeezing, pinching, and/or shearing effect on the congruency process. This is because of
the tendency of such methodologies to try to impose a structural character onto an aspect
of reality that does not really fit.
This attempt
to force-fit reality into preconceived categories of whatever description, causes the
hermeneutic of the phenomenology of the experiential field to develop wrinkles, bumps,
lacunae, and so on. These get in the way of achieving a complete congruency relationship
or merging of horizons.
One might
suppose that when various dimensional manifolds are brought into contact with one another,
the complex structural character of these dimensions may often prevent them from making
contact at every 'point' of their respective manifolds. In this respect, dimensional
manifolds are somewhat like material surfaces.
Moreover,
like material surfaces, the character of the interaction between dimensional manifolds may
be shaped by the character of the fractal properties of the interacting dimensions. Such
fractal properties are, in turn, an expression of the spectrum of constraints and degrees
of freedom through which the order-field establishes the structural character of the
various dimensions involved in the interaction.
In line with
the foregoing, one might treat the horizon as a manifold of complex structural character
which is formed by the interaction of a number of different dimensions. One, then, could
construe the notion of a merging of horizons as a fractal like problem involving the
interaction or dialectic of manifold latticeworks in n-dimensions. Thus, the fact that
hermeneutical structures may not coincide with the ontological structures to which the
former are making identifying reference could be conceived of as a function of certain
incongruencies that occur along the horizon linking the respective fractal characters of
ontological events and hermeneutical activity.
The problem of turbulence
The fractal
perspective is rooted in the assumption that beneath all the discontinuity, irregularity
and fragmentation lies a symmetry or invariance governing how such phenomena organize
themselves around self-similar themes across various levels of scale. Fractal geometry
represents a means of trying to establish a link between chaotic behavior and ordered
behavior. It is a means of trying to show or suggest why one could find order in the midst
of chaos, as well as chaos in the middle of order.
Furthermore,
fractal geometry is an attempt to account for how these pockets of chaos and order are
linked by a set of symmetry themes that would lead to the emergence of the same
juxtaposition ing of chaos and order on any and every level of scale one cared to examine.
However, as
Mandelbrot, himself, has admitted, although fractal geometry provided a useful descriptive
tool, it often fell short of being able to answer a number of fundamental questions.
For example,
his theory could not answer why, or how, the juxtaposition ing of order and chaos is
possible. He also could not account for why, or how, symmetry was able to be preserved
across various levels of scale, despite the presence of destabilizing forces and
fluctuations and perturbations in the system.
Turbulence
has been described as a sort of a breakdown of laminar or smooth flow across all levels of
scale in a given system. Under normal circumstances, when fluctuations arise
in a laminar system, these fluctuations tend to disappear or die out.
However,
when some critical point of intensity and/or number of fluctuations has been crossed, the
system tends to destabilize in a catastrophic manner. In other words, turbulence occurs.
Turbulence
disrupts the flow of energy in a system. Therefore, turbulence impedes the character of
the dynamics or motion normally governing a system. Pockets of turbulence both divert
energy away from the rest of the system, as well as constitute sources of drag for the
normal paths of motion within the system.
Trying to
discover how the transition from a laminar flow to a turbulent flow occurs has long been a
problem in a variety of sciences. Unfortunately, whatever success scientists have had in
coming to grips with this problem, has been limited to descriptive approximations about particular
situations. Scientists have not had much success in providing an account that
incorporates a set of universal principles capable of explaining (and not just
predicting or describing), in precise mathematical formulation, why turbulence occurs in
systems previously characterized by laminar flow.
One of the
assumptions traditionally made about turbulence is that the disturbances are distributed
uniformly throughout a system. Thus, some scientists approached turbulence in
terms of a model that described the perturbations arising in a given system as a sort of
homogeneous phenomenon.
However,
subsequent work has shown that the set of vorticies making up turbulence tend to be
unevenly and intermittently distributed in a system. In fact, scientists have discovered
that when one examines any given vortex of perturbation in finer detail, the vortex itself
breaks down into an intermittent pattern of laminar and turbulent motion.
The standard
account of the transition problem usually is expressed in terms of some variation on the
account originally provided by the Russian scientist, Lev D. Landau. Landau believed any
given system of fluid motion consisted of a coupling of frequency components that were a
function of the energy in the system. As new energy was fed into the system, new
frequencies emerged in the system one at a time.
Yet, these
frequencies were not independent of one another. They were tied to the character of
neighboring frequency patterns.
Consequently,
there were only a limited number of degrees of freedom which could be realized in such a
coupled system. In other words, the potential for complex, autonomous frequency components
arising in a system of fluid motion is curtailed by the dampening effect which the
vectoring of neighboring frequencies has upon new energy components being introduced into
the system.
On some
occasions, for unknown reasons, an influx of energy introduced, into the system leads to a
series of unstable motions which are not dampened by neighboring frequency patterns.
According to Landau, these unstable motions tend to accumulate or hang together. As a
result, the amalgamation of unstable motions creates complex frequency structures
comprised of a set of overlapping frequency patterns of different rhythms, speeds and
sizes.
While this
model appeared to fit the overall characteristics of turbulent phenomena, it was virtually
useless in helping one to understand how turbulence actually arose. Moreover, Landau's
model did not provide one with a means of precisely determining either: (a) when an influx
of energy would lead to the appearance of a new frequency in the system, or (b) what the
value of that frequency would be if it were to arise.
In short,
the increase of one or more vectors leads to a catastrophic and discontinuous change in
the macroscopic properties of the system. Significantly, there is only a slight difference
in the average energy displayed by a system between a point just prior to the critical
transition juncture and the actual point of transition itself.
However,
suddenly, the macroscopic characteristics of the system are being regulated by laws. Such
laws could not have been anticipated on the basis of knowledge of the microscopic
properties of the system prior to reaching the critical phase transition point.
Catastrophic transitions and education
The
foregoing sort of subtle shift in average energy past some critical level that is
subsequently followed by a sudden, discontinuous alteration in system properties, seems
like the abrupt transition which occurs in relation to the Necker cube illusion, when a
slight change in focal/horizonal interaction takes place. This focal/horizonal dialectic
can be altered in marginal ways until it reaches some critical juncture, beyond which the
perspective goes through a catastrophic and discontinuous change.
In fact, any
latticework or set of interacting latticeworks will have one or more critical values
inherent in the structure's spectrum of ratios of constraints and degrees of freedom. When
these values are exceeded, Necker-like transitions occur.
These
transitions, however, are not continuous in any traditional mathematical sense, but are
more akin to the way the discrete runners in a relay race keep the process continuous by
handing off the baton to one another. As such, the Necker-like alteration does not go
through every intermediate point between the pre-critical structural character and the
post-critical structural character - but, rather, at different junctures, one process
leaves off and another one begins.
There are a
number of intriguing questions, issues and problems which arise when one reflects on the
issue of sudden shifts in hermeneutical phase transitions in the context of education. For
example, one needs to determine whether certain kinds of vectoring (in the form of
teaching, curriculum, textbooks and so on) consistently will lead to certain sorts of
catastrophic changes of understanding, behavior and so on.
There is
also the problem of determining whether educational changes can be brought about in a
non-catastrophic manner. Must one suppose that sudden, discontinuous changes are an
intrinsic feature of all learning situations? Or, looked at from another perspective, one
might ask: Does learning which is rooted in catastrophic or sudden phase transitions have
greater heuristic value than does learning which is rooted in non-catastrophic phase
transitions?
An
additional issue revolves about the question of whether or not one must individualize
education because different people will have different kinds of critical points of
catastrophic phase transition. Alternatively, despite differences from one individual to
the next, could one suppose there is sufficient self-similarity to be observed across a
group of individuals that one does not need to individualize education in the foregoing
sense?
Finally, one
might seek to determine if there are links between indoctrination and chaos theory. For
instance, one might treat indoctrination as the active, or even passive, attempt, whether
intended or not, to prevent the individual from reaching certain kinds of critical points
of phase transition during the course of the life-cycle. These critical points might cover
a whole host of developmental issues, ranging from emotional issues, to political, social,
economic, intellectual, creative, and spiritual issues.
Fixed point limit cycle and chaotic attractors
Traditionally,
there have been two kinds of attractors used to describe the dynamics of phase space.
These are: (a) the fixed point attractor and (b) the limit cycle. The fixed point
attractor tends toward a single form of steady state. The limit cycle attractor gravitates
toward a continuously repeating, oscillating structural form. The character of this
oscillating structural form will depend on the vectoral forces at work in the system under
consideration.
Phase space
is a means of giving visual representation to central themes of complicated systems.
Essentially, one uses the movement of a point to describe the dynamics of certain thematic
aspects of a given system over time. The point constitutes the intersection of two or more
co-ordinates, with the number of co-ordinates depending on the number of variables on
which one is trying to keep tabs. Each independent variable constitutes a degree of
freedom and is represented as another dimension in phase space.
When a new
point is plotted, this represents the changing relationship between, or among, the
variables being studied. The curve or geometric figure described by a series of plotted
points gives expression to the dynamics of the system over time.
For example,
consider a phase space describing the relationship of two variables, velocity and
position, of a moving pendulum. The curve described by plotting the relationship between
velocity and position over time is a loop.
Adding
energy to the system, by permitting the pendulum to cover a greater arc and at a faster
rate, or withdrawing energy from the system (as would be the case with a pendulum which
covers less distance in its moment and does so at a slower rate), will not change the fact
that the dynamics of either kind of system will still be described by a loop. The only
difference will be in the size of the loop.
In general,
the more energy associated with the movement of the pendulum, the larger will be the size
of the loop which is plotted. On the other hand, the less energy contained in a pendulum's
movement, the smaller will be the size of the loop being plotted to describe the dynamics
of such a system.
If one
introduces friction as a third variable into the above system, this will be a source of
drag. The effect of the drag will be to dissipate the energy contained in the pendulum's
movement. As more and more energy is drained from the pendulum's movement, due to the
effect of friction, the loop describing the dynamics of the system will become smaller and
smaller.
Friction
acts as a fixed point attractor. The loop describing the dynamics of the pendulum system
shrinks, reflecting the presence of friction. Eventually, the loop is drawn toward
equilibrium where position is fixed and velocity is zero.
Therefore,
in the phase space describing a pendulum system, dissipation of energy is shown by the way
the loop representing the dynamics of that space gravitates toward some central, fixed
point. The contraction of a figure in phase space represents the dissipation of energy in
a system as the variables of that system are drawn toward an attractor of some sort which
is constraining the way the degrees of freedom are manifesting themselves.
When
turbulence occurs, energy is both flowing into, as well as being dissipated out of, the
system. As a result, the dynamics of a system beset by turbulence do not tend toward any
point of equilibrium. Therefore, one cannot use the idea of fixed point attractor to
describe what goes on in the midst of such turbulence.
The only
other kind of attractor traditionally used to describe the dynamics of phase space is the
limit cycle. In the limit cycle attractor one has a rather special orbital loop giving
expression to the movement of a point that describes the changing relationship between, or
among, a set of variables.
This orbital
loop tends to attract all other orbital loops which might appear in the system. Thus,
there is one orbital loop that constitutes a limit toward which other loops in the system
will gravitate.
Unlike a
fixed-point attractor, however, although the limit cycle displays equilibrium or
stability, it does not tend toward a fixed, zero, energy point. A limit cycle describes a
periodic dynamic and, therefore, repeats itself in a regular way.
Sometimes a
given phase space may be characterized by several attractors. For example, a given system
may have both a fixed point attractor component as well as a limit cycle attractor
component. Under such circumstances, each attractor component has its own basin, and each
basin has a shaping influence on the structural character of the system in which it
exists.
Although any
given point in phase space represents a possible dynamical state of that space, in point
of fact, the long term structural character of a phase space is completely described by
the kind of attractor to which such a phase space is drawn. Any kind of motion deviating
from the long term structural tendencies of a given system governed by a fixed-point or
limit cycle, will be nothing more than a fleeting fluctuation.
These
fluctuations will die out in time. In short, attractors embody the property of stability
in the sense that the dynamics of a given phase space tend to gravitate toward the form of
the attractor which is governing that phase space.
Turbulent
systems. however, present a problem for traditional modes of phase space analysis. The
very nature of turbulence is that it doesn't give expression to any single rhythm. It
embraces a whole spectrum or range of rhythms which dialectically interact to produce the
complex structural character of turbulence.
In 1971,
David Ruelle, a mathematical physicist, and Floris Takens, a Dutch mathematician, claimed
that turbulence must be described in terms of a special kind of attractor.
Like fixed point and limit cycle attractors, this new kind of attractor would show
stability.
However,
unlike either of the two traditional forms of attractor, the new form of attractor would
be nonperiodic. Thus, it would not repeat any given rhythmic sequence. A further feature
of the new sort of attractor being proposed was that despite not repeating any cyclical
pattern, the differences between one cycle and another would manifest variations of but a
few degrees of freedom.
According to
Ruelle and Takens, the dynamics of turbulence could be described in phase space by the
interaction of only a small number of vector variables. The interaction of such variables
could be described by plotting a series of points that constitute the intersection of a
small set of co-ordinate axes or dimensions. Thus, low-dimensionality was a further
property of the new kind of attractor being introduced.
In effect,
the strange attractor (also known as a chaotic attractor) being proposed by Ruelle and
Takens already existed in the form of the Lorenz attractor. The Lorenz
attractor possessed the necessary properties of being stable and nonperiodic, yet showing
low-dimensionality.
Furthermore,
the loops of the Lorenz attractor never repeated themselves, nor did they intersect
themselves. Nevertheless, the Lorenz loops gave expression to this variety within a finite
envelope of space.
The structural character of hermeneutical attractors
Any ratio of
constraints and degrees of freedom gives expression to an attractor. The dialectical
character of such a ratio determines the properties of the attractor basin or sphere of
influence arising as a manifestation of the attractor.
Therefore,
hermeneutical structures (which can be construed in terms of a complex dialectic of
various spectrums of ratios of constraints and degrees of freedom) give expression to
attractors and, therefore, attractor basins. Some hermeneutical structures form
fixed-point structures. Other hermeneutical structures form limit cycle attractors, while
still other such structures form chaotic attractors.
In general
terms, there is a dynamic dialectic occurring along the boundaries linking two or more
hermeneutical attractor systems. Each attractor has a basin which serves to shape and
orient the forces characteristic of that attractor. The basin gives expression to the
hermeutical counterparts to vectoral and tensoral components which establish the
parameters marking the outer limits of the hermeneutical attractor's sphere of influence.
As indicated
earlier, not all dynamical systems are governed by just one state of equilibrium. Some
systems have two equilibrium states, and others may have more than two states of
equilibrium. This is especially true in the case of hermeneutical systems.
Each
equilibrium state constitutes an attractor, and each attractor gives expression to a set
of boundary properties. Where two or more attractors come together, the boundary
separating them can be ( but may not be) both complicated and turbulent.
Moreover,
even though the long-term character of such dialectical interaction might not be chaotic,
chaotic properties may surface along the boundary regions separating one hermeneutical
attractor basin from another. As a result, predicting in which direction the system will
go can become extremely difficult.
The study of
hermeneutical, attractor, fractal, basin boundaries is, like its counterpart in nonlinear
dynamics, concerned with the phase transitions occurring at certain threshold values along
the boundaries of interacting hermeneutical basin attractors. This occurs as one goes from
laminar flow to catastrophic behavior, to a, final, non-chaotic equilibrium state within
such systems.
In a sense,
constraints and degrees of freedom have a sort of yin and yang relationship. In other
words, there are degrees of freedom within any given set of constraints, just as there are
constraints within any given set of degrees of freedom.
In light of
the foregoing comments, one cannot really separate the ratio of constraints and degrees of
freedom. The integrity of a latticework's structural character requires both.
Indeed, the
yin/yang relationship of constraints and degrees of freedom is somewhat reminiscent of the
relationship between information and noise which Mandelbrot discovered in relation to
messages communicated over telephone lines. As a result, irrespective of the level of
scale through which one engages a given structure, there will be a ratio of constraints
and degrees of freedom that gives expression to the character of that structure. This
ratio serves as a signature for a given structure.
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