Deterministic Ideas In A Chaotic Milieu
N. Lygeros - J.D. Martinez
 
 
 

The fact that a disordered complex structure, that is apparently random
in nature, allows an ordered substructure is not the result of Chaos
Theory but
rather of Ramsey's Theory. In a most explicit manner the latter theory
asserts that there lies a substructure possessing a given property
within all
sufficiently large structures. It is clear that the sufficient condition
regarding the structure's height is always fulfilled within the
situations where Chaos
Theory is applicable.

Chaotic behaviour within a system displays delicate sensitivity to tiny
changes in such a manner that any ignorance of its present state leads
to complete
ignorance concerning its state after a brief period of time, for
example, weather prediction is affected by this problem. Order develops
on a large scale
through the concatenation of several small-scale events on the verge of
instability. In 1961 the metereologist Lorenz accidentally discovered
the butterfly
effect, whereby the movement of a butterfly's wing(s) taking place today
in one part of the world brings about an extremely small change in the
state of
the atmosphere which can have radical repercussions on global weather
patterns.

In Lorenz's words; the butterfly effect is the phenomenon that a small
alteration in the state (the condition of a system at one instant) of a
dynamical or
deterministic system (a system in which later states evolve from
previous ones according to a fixed law) will cause subsequent states to
differ greatly
from the states that would have followed without the alteration. Lorenz
referred to this cumulative effect as sensitive dependence. After a
certain length
of time the atmosphere's behaviour diverges from the expected behaviour.
Lorenz published his seminal article in 1963, wherein he referred to the

possibility of long-term weather forecasting via the prediction or
estimation of its periodic variations. Lorenz lent his name to the
Lorenz Attractor: in
other words,chaotic motion in a dissipative(volume-decreasing) system.

Currently, it is possible to perform a conveyance of frame and to apply
this approach to the generation of an idea in the brain or encephalon.
Within this
materialistic context, where thought is considered to be like an
emergence of cerebral activity, each idea would involve the activation
of a series of
neurones that constitute a fractal trajectory within the topological
space represented by the brain.

The simplest form of 'fractal' objects (a concept introduced by Benoit
Mandelbrot in 1975) are self-similar or self-affine. In other words,
these fractals do
not change their appearance when viewed under a microscope of arbitrary
magnifying power. Natural boundaries such as coast lines apparently
become
longer the finer the scale on which we measure them. One of the
characteristics of the boundary is its self-similarity, also known as
Julia Sets. In 1980
Mandelbrot discovered what was later to be termed the Mandelbrot Set,
with its associated spiral-like peninsula on its edge. The term
'multifractality'
was first coined by Mandelbrot. Therefore, it would be theoretically
possible to attribute a fractal dimension to an idea in order to
distinguish it from white
noise. Consequently, within this framework thought would be made up of a
succession of deterministic ideas developing within a chaotic
environment or
milieu.

The computation of the fractal dimension of an irregular phenomenon such
as an idea (within consecutive bursts of cerebral activity arising due
to
successive deterministic synaptic-neuronal activation processes) depends
on several factors. One critical factor that comes into play and that
stands out
over and above the background white noise concerns the competing
neurotransmitted signals within the chaotic, quasi-random milieu of the
mind. The
human brain's potential electrical and neurochemical transmission relies
heavily upon "encephalic goodness", i.e. neuronal efficiency, to the
extent that a
finite number of cerebral signals will trigger off appropriate brain
responses during the complex process of thought generation, hence giving
rise to the
emergent property of human conscience and awareness.

First of all, it is necessary to bring to light the fact that the
deterministic view can indeed be identified as it is the brain's normal
functioning mode. So,
without doubt, in order to resolve this problem and therby extract the
information concerning an idea's fractal dimension it will be necessary
to make use
of the wavelet multifractal approach. In the end, the study of the
fractal spectrum, obtained with the aid of wavelets, will enable the
separation which is
associated with human thought.

Human thought can be abstracted as the end product of a finite series of
"stochastic" processes involving very swift and oscillating,
bidirectional
exchanges between cerebral areas of low and high entropy, thereby
epitomizing the idea of structure or order within chaos.The fractal
dimensionality of
thought, as derived from Chaos Theory, is postulated on the basis of the
nesting of an apparently random set of cerebral events within the
ordered
framework of a neurologically and topologically defined, finite brain
architecture. There exists an 'apparent random' connection of a huge
number of
neurones creating an almost infinite number of possible permutations and
combinations, but this is all based on a few 'rules' laid down by the
human
genetic code that manages to govern how body cells should connect and
interact.

The following is a paradigm of a deterministic structure comprising a
nondeterministic substructure : the prime numbers and the classes 4n+3
and 4n+1.
The set of prime numbers is deterministic in the sense that via
elliptical curves one can give a certification of the primality of a
number. In addition, if we
consider the difference between the numbers of prime numbers of the
class 4n+3 and those of 4n+1, we can demonstrate that it changes sign an
infinite
number of times. Nevertheless, it remains positive over extremely long
ranges. It is the sign of this difference that constitutes a
nondeterministic object
within the deterministic structure of prime numbers.

This paradigm shows that the nesting of determinism and indeterminism is
dual. Chaos and order can be found to coexist in a type of competitive
symbiosis. This result is altogether typical within the framework of a
fractal mentality since it highlights the complexity of the notion of
boundary.
Complex adaptive systems thrive in the hinterland between the
inflexibilities of classical determinism and the vagaries of chaos. The
theory of Laplace's
Classical Determinism can be applied to diverse chaotic systems such as;
a snowflake, a time series with its random walk, a pinball machine, and
Hyperion's almost random oscillations along its own axes with respect to
its eccentric, elliptical orbit around Saturn. These stable systems
allow stable
predictions to be made. Thought pattern generation is unlikely to be a
phenomenon involving similar stability in terms of experimental test and
retest
verification.

The aforegoing consolidates the idea that the interaction between
phenomena of low and high entropy is not a problem but rather a reality
which should be
taken into account. It can be concluded that the specifics of the
qualitative and quantitative interactions that take place within the
brain and along the
brainstem still remain to be elucidated.

Recommended Bibliography.

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1990.
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Applications. Cambridge: Cambridge University Press, 1985.
Gleick,J.: Chaos : Making a New Science. New York: Viking Press, 1987.
Guckenheimer,J.,and Holmes,P.: Nonlinear Oscillations, Dynamical
Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag,
1986.
Hofstadter,D.: Metamagical Themas. New York: Basic Books, 1985.
Julia,G.: Oeuvres de Gaston Julia. Paris: Gauthier-Villars, 1968.
Kadanoff,L.P.: From Order to Chaos. Singapore: World Scientific
Publishing Co., 1993.
Lasota,A.,and Mackey,M.C.: Chaos, Fractals, and Noise. Stochastic
Aspects of Dynamics. New York: Springer-Verlag, 1994.
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Lorenz,E.N.: Deterministic Nonperiodic Flow. Journal of the Atmospheric
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Massopust,P.R.: Fractal Functions, Fractal Surfaces, and Wavelets. San
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Peitgen,H.-O.,and Richter,P.H.: The Beauty of Fractals. New York:
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Peitgen,H.-O.,and Richter,P.H.: The Science of Fractal Images. New York
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Schroeder,M.: Fractals, Chaos, Power Laws. Minutes from an Infinite
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Schuster,H.G.: Deterministic Chaos. Weinheim: Physik-Verlag GmbH, 1984.
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Stewart,I.: Does God play dice? The New Mathematics of Chaos. Penguin
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Tvede,L.: Business Cycles. The Business Cycle Problem from John Law to
Chaos Theory. Netherlands : Harwood Academic Publishers GmbH, 1997.
 

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