## 142 - Deterministic Ideas In A Chaotic Milieu. (with J. Martinez).

### J. Martinez, N. Lygeros

The fact that a disordered complex structure, that is apparentlyrandom in nature, allows an ordered substructure is notthe result of Chaos Theory but rather of Ramsey’s Theory. In a mostexplicit manner the latter theory asserts that there lies a substructurepossessing a given property within all sufficiently large structures. Itis clear that the sufficient condition regarding the structure’s heightis always fulfilled within the situations where Chaos Theory isapplicable.

Chaotic behaviour within a system displays delicate sensitivity totiny changes in such a manner that any ignorance of its present stateleads to complete ignorance concerning its state after a brief period oftime, for example, weather prediction is affected by this problem. Orderdevelops on a large scale through the concatenation of severalsmall-scale events on the verge of instability. In 1961 the metereologistLorenz accidentally discovered the butterfly effect, whereby themovement of a butterfly’s wing(s) taking place today in one part of theworld brings about an extremely small change in the state of theatmosphere which can have radical repercussions on global weatherpatterns.

In Lorenz’s words; the butterfly effect is the phenomenon that asmall alteration in the state (the condition of a system at one instant)of a dynamical or deterministic system (a system in which later statesevolve from previous ones according to a fixed law) will cause subsequentstates to differ greatly from the states that would have followed withoutthe alteration. Lorenz referred to this cumulative effect as sensitivedependence. After a certain length of time the atmosphere’s behaviourdiverges from the expected behaviour. Lorenz published his seminalarticle in 1963, wherein he referred to the possibility of long-termweather forecasting via the prediction or estimation of its periodicvariations. Lorenz lent his name to the Lorenz Attractor: in otherwords,chaotic motion in a dissipative(volume-decreasing) system.

Currently, it is possible to perform a conveyance of frame and toapply this approach to the generation of an idea in the brain orencephalon. Within this materialistic context, where thought isconsidered to be like an emergence of cerebral activity, each idea wouldinvolve the activation of a series of neurones that constitute a fractaltrajectory within the topological space represented by the brain.

The simplest form of ‘fractal’ objects (aconcept introduced by Benoit Mandelbrot in 1975) are self-similar orself-affine. In other words, these fractals do notchange their appearance when viewed under a microscope of arbitrarymagnifying power. Natural boundaries such as coast lines apparentlybecome longer the finer the scale on which we measure them. One of thecharacteristics of the boundary is its self-similarity, also known asJulia Sets. In 1980 Mandelbrot discovered what was later to be termed theMandelbrot Set, with its associated spiral-like peninsula on its edge.The term ‘multifractality’ was first coined by Mandelbrot. Therefore, itwould be theoretically possible to attribute a fractal dimension to anidea in order to distinguish it from white noise. Consequently, withinthis framework thought would be made up of a succession of deterministicideas developing within a chaotic environment or milieu.

The computation of the fractal dimension of an irregularphenomenon such as an idea (within consecutive bursts of cerebralactivity arising due to successive deterministic synaptic-neuronalactivation processes) depends on several factors. One critical factorthat comes into play and that stands out over and above the backgroundwhite noise concerns the competing neurotransmitted signals within thechaotic, quasi-random milieu of the mind. The human brain’s potentialelectrical and neurochemical transmission relies heavily upon “encephalicgoodness”, i.e. neuronal efficiency, to the extent that a finite numberof cerebral signals will trigger off appropriate brain responses duringthe complex process of thought generation, hence giving rise to theemergent property of human conscience and awareness.

First of all, it is necessary to bring to light the fact thatthe deterministic view can indeed be identified as it is the brain’s normalfunctioning mode. So, without doubt, in order to resolve this problem andtherby extract the information concerning an idea’s fractal dimension itwill be necessary to make use of the wavelet multifractal approach. Inthe end, the study of the fractal spectrum, obtained with the aid ofwavelets, will enable the separation which is associated with humanthought.

Human thought can be abstracted as the end product of a finiteseries 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.Thefractal dimensionality of thought, as derived from Chaos Theory, ispostulated on the basis of the nesting of an apparently random set ofcerebral events within the ordered framework of a neurologically andtopologically defined, finite brain architecture. There exists an’apparent random’ connection of a huge number of neurones creating analmost infinite number of possible permutations and combinations, butthis is all based on a few ‘rules’ laid down by the human genetic codethat manages to govern how body cells should connect andinteract.

The following is a paradigm of a deterministic structurecomprising a nondeterministic substructure : the prime numbers and theclasses 4n+3 and 4n+1. The set of prime numbers is deterministic in thesense that via elliptical curves one can give acertification of the primality of a number. Inaddition, if we consider the difference between the numbers of primenumbers of the class 4n+3 and those of 4n+1, we can demonstrate that itchanges sign an infinite number of times. Nevertheless, it remainspositive over extremely long ranges. It is the sign of this differencethat constitutes a nondeterministic object within the deterministicstructure of prime numbers.

This paradigm shows that the nesting of determinism andindeterminism is dual. Chaos and order can be found to coexist in a typeof competitive symbiosis. This result is altogether typical within theframework of a fractal mentality since it highlights the complexity ofthe notion of boundary. Complex adaptive systems thrive in the hinterlandbetween the inflexibilities of classical determinism and the vagaries ofchaos. The theory of Laplace’s Classical Determinism can be applied todiverse chaotic systems such as; a snowflake, a time series with itsrandom walk, a pinball machine, and Hyperion’s almost random oscillationsalong its own axes with respect to its eccentric, elliptical orbit aroundSaturn. These stable systems allow stable predictions to be made.Thought pattern generation is unlikely to be a phenomenon involvingsimilar stability in terms of experimental test and retest verification.

The aforegoing consolidates the idea that the interaction betweenphenomena of low and high entropy is not a problem but rather a realitywhich should be taken into account. It can be concluded that thespecifics of the qualitative and quantitative interactions that takeplace within the brain and along the brainstem still remain to beelucidated.

Recommended Bibliography.

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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, andBifurcations 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, andNoise. 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 theAtmospheric 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 ofFractals. New York: Springer- Verlag, 1986.

Peitgen,H.-O.,and Richter,P.H.: The Science of FractalImages. 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 BooksLtd, Harmondsworth, 1989.

Tvede,L.: Business Cycles. The Business Cycle Problem from John Law toChaos Theory. Netherlands : Harwood Academic Publishers GmbH, 1997.