Deterministic Ideas In A Chaotic Milieu

J. Martinez, N. Lygeros




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.

Barnsley,M.F.: Fractals Everywhere. Boston: Academic Press, 1988.
Devaney,R.L.: Chaos, Fractals, and Dynamics. Boston: Addison-Wesley, 1990.
Falconer,K.J.: Fractal Geometry. Mathematical Foundations and 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.
Lorenz,E.N.: The Essence of Chaos. Washington : University of Washington Press, 1993.
Lorenz,E.N.: Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences, 20 (1963): 130-141.
Mandelbrot,B.B.: Fractals: Form, Chance and Dimension. San Francisco: Freeman, 1977.
Mandelbrot,B.B.: The Fractal Geometry of Nature. San Francisco: Freeman, 1982.
Massopust,P.R.: Fractal Functions, Fractal Surfaces, and Wavelets. San Diego,California: Academic Press, 1994.
Peitgen,H.-O.,and Richter,P.H.: The Beauty of Fractals. New York: Springer- Verlag, 1986.
Peitgen,H.-O.,and Richter,P.H.: The Science of Fractal Images. New York Springer-Verlag, 1988.
Pickover,C.: Computers, Pattern, Chaos and Beauty. New York: Sutton, 1990.
Schroeder,M.: Fractals, Chaos, Power Laws. Minutes from an Infinite Paradise. New York: Freeman, 1991.
Schuster,H.G.: Deterministic Chaos. Weinheim: Physik-Verlag GmbH, 1984.
Schwenk,T.: Sensitive Chaos. New York Schocken Books, 1976.
Stewart,I.: Does God play dice? The New Mathematics of Chaos. Penguin Books Ltd, Harmondsworth, 1989.
Tvede,L.: Business Cycles. The Business Cycle Problem from John Law to Chaos Theory. Netherlands : Harwood Academic Publishers GmbH, 1997.







free counters


Opus