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Vladimir Dimitrov

Chaos and Complexity theory studies nonlinear processes: Chaos explores how complexly interwoven patterns of behaviour can emerge out of relatively simply-to-describe nonlinear dynamics, while Complexity tries to understand how relatively simply-to-describe patterns can emerge out of complexly interwoven dynamics.

Both Chaos and Complexity build the core of Complexity Science actively promoted in the research publications of the Santa Fe Institute in USA established in 1984. This institute draws scientists from universities and research institutions throughout the world - mathematicians and computer scientists, economists and social scientists, biologists and ecologists, psychologists and philosophers, interested in exploring the underlying dynamic patterns and regularities behind a wide spectrum of complex and chaotic phenomena and processes. The aim of the inquiry is to help define new research directions and to shed light on real-life problems and enigmas that challenge our global society today. 

A Historical Outlook
Chaos and Complexity emerge from non-linear mathematics. Theoretical and applied development of mathematical cybernetics and computer science made it possible for many mathematicians and physicists to step out of the framework of linearity, continuity and smoothness, and to approach problems belonging to the world of non-linearity, discontinuity and transformations.

With their pioneering works on local stability (instability) of dynamical systems in the last decade of the 19 century, the Russian mathematicians Andrey Lyapunov and Sophia Kovalevskaya are viewed as the founders of the single and most creative and prolific stand of thought in the analysis of dynamic discontinuities and non-linearities up to the present day, the Russian School. Significant successors to Lyapunov and Kovalevskaya include A. Andropov and L. Pontryagin (1937) who crucially advanced the theory of structural stability, A. Kolmagorov (1941) who developed foundation of the mathematical theory of turbulence (chaotic fluid dynamics) and V. Arnold (1968) with his most completely classification of mathematical singularities (catastrophes).

In 1962 Kolmagorov and Obukhov mathematically demonstrated possibility of an intermittency of chaotic fluid dynamics and emergence of patterns of order out of a turbulent flow.

The famous KAM theorem of Kolmagorov-Arnold-Moser (1978) threw light over the unresolved 3-body problem of Laplacean-Newtonian celestial mechanics - a problem firstly approached by the French mathematician Henri Poincaré (1890) who, facing its unsurmountable computational difficulty, saw the possibility of existence of a non-wandering (dynamically stable) solution of extreme complexity, and thus firstly predicted the existence of an attractor in chaotic dynamics. According to the KAM theorem, the trajectories studied in classical mechanics are neither completely regular nor completely irregular, but they depend very sensitively on the chosen initial states: tiny fluctuations can cause chaotic development.

The sensitive dependence of initial conditions as a true sign of chaotic dynamics was firstly studied mathematically by E. Lorenz (1963) in his three-equation model of atmospheric flow. A slight change in parameter value of the model had generated significantly different behaviour. Lorenz labelled this phenomenon as "butterfly effect". He found also that dynamic trajectories described by the equations moved very quickly along a branched, S-shaped, two-dimensional attractor with of a ëstrangeí form.

The term ëstrange attractorí was firstly used by D. Ruelle and F. Takens (1971) when describing dynamic behaviour of their mathematical model of fluid turbulence.

In 1975 the American mathematicians Li and Yorke firstly used the term "chaos" in their widely cited paper "Period three implies chaos" (Amer. Math. Monthly 82, 985) to characterize the emergence of an uncountably infinite number of asymptotically aperiodic orbits in system dynamics. This was not a new theorem - Li and Yorkeís result was already proven (11 years earlier) in a Russian School paper written by Sarkovski.

In 1972 the French mathematician R. Thom introduced the Catastrophe Theory to describe topologically discontinuities (topology is the mathematical study of properties of objects which are preserved through deformations, twistings, and stretchings) appearing in a wide variety of phenomena in many different disciplines and contexts; he identified seven elementary catastrophes emerging in system dynamics characterised by four control variables. Later, the mathematicians from the Russian school - Arnold, Gusein-Zadeh and Varchenko proved that for higher level of dimensionality the number of possible catastrophes become infinite and hence very difficult to classify. This is where the realm of chaos and fractal geometry begins.

Fractal Geometry was introduced by Benoit Mandelbrot in his famous book "Fractal Geometry of Nature"(1983). For Mandelbrot, the closer one looks at nature, the more irregular and chaotic it becomes. Fractals have self-similar structure. They are infinitely complex - the closer one looks at a fractal, the more detail one sees. According to Mandelbrot, chaotic dynamics crucially depend on the existence of a fractal attractor, that is, a strange attractor with fractal dimensionality. Other chaoticians, like Eckmann and Ruelle have argued that it is the sensitive dependence on initial conditions which is the true meaning of chaos and not the existence of a fractal attractor - chaotic dynamics may occur even without a fractal attractor.

The bridge from Chaos to Complexity passes through the studies of self-organization processes in complex cybernetic systems by A.Ivaknenko, Yu. Zaichenko and V. Dimitrov (1976) from the Ukrainian Institute of Cybernetics in Kiev, where the Group Method of Data Handling has been developed ( By modelling some fundamental principles of self-organization of living systems, this methods offered practical algorithms for solving complex ecological, economical and engineering problems related to pattern-recognition, classification, scenarios generation, etc.

Self-organization behaviour can be exhibited by far-from-equilibrium chemical systems as it was shown by the Nobel-prize winner (1977) in chemistry Ilya Prigogine. According to the results of his studies, the inorganic chemical systems can exist in highly non-equilibrium conditions impregnated with a potential for emergence of self-organizing chemical structures. The more complex the aggregation of these structures, the stronger the tendency for macro-molecules to ëorganizeí themselves.

Significant contribution in understanding self-organization process in complex economical and biological systems have been made by B. Arthur (1990) and S. Kaufman (1992), respectively. They both work with the Santa-Fe institute for complexity studies.

Major Discoveries of Chaos and Complexity: A Social Perspective
Several stunning discoveries of the theories of Chaos and Complexity shattered the logical foundations of science built over the span of many centuries:

I. First discovery: Prediction and determinism are incompatible: we cannot predict long-term beahaviour of complex systems, even if know their precise mathematical description.

This discovery brought to an end the ambitious dreams of science and technology about the omnipotent power of human reason to solve all the complex problems of this world.

The reason why we cannot say much about a complex dynamic is because of its enormous sensitivity: even an infinitely small change in the starting conditions of a complex process can result in drastically different future developmens (Lorenzí butterfly effect).

The butterfly effect has no place in linear systems: in those systems the changes of output variables are proportional to the changes of input variables. For example, we put a bit more sugar in the tea, and the tea become sweeter; if then we add a bit of hot water, the sweetness decreases in an exact proportion to the added water. Science prefers to work with linear systems - they are easily predictable.

Unfortunately, real-life is non-linear. The variables which we use to describe real-life complexity have their critical (or threshold) values - if we ëpushí a complex system beyond these values, even slightly, huge changes may occur with the system as a whole. If we are standing at the edge of a cliff, a tiny bit of movement ahead could demonstrate the butterfly effect of Lorenz in a tragic and irreversible way. Although in this case we still have an idea of what may happen with us, in complex situations non-linearity usually leads to unpredictable beahaviour. Weather, the stock-market, human and environmental health, national economies, large business or socio-political formations are examples of unpredictable non-linear systems.

When we deal with complex social issues, we cannot rely too much upon the long-term predictions of the experts. Tiny changes in the conditions, under wich their predictions have been made, alter the prediction significantly.

We only waste time and energy when trying to force complex system development in a pre-planned, non-negotiable direction. Much wiser is to learn how to nudge from within the system, how to manage and guide from inside its chaotic dynamics.

Unpredictability of complex behaviour is not an obstacle on the way to grasp this behaviour. On the contrary, by exploring unpredictable dynamics of non-linear processes, we can gain insights with enormous predictive power. Here are some insights helpful in undersatnding and working with complexity.

(1) There are no negligible actions: even randomly chosen and seemingly insignificant actions can lock-in, accelerate and amplify, beyond our ability to control their future directions; because of this effect, complex systems are permanently driven out of equilibrium.

What does this mean for practitioners in the field of human and environmental health? Open minds and hearts to see and feel the subtleties of the system of human and environmental health as a whole - these subtelties could be crucial for understanding changes in its interwoven dynamics.

(2) In contrast to equilibrium physics and mechanics, where a critical state is an exception, in far-from-equilibrium reality (life) critical state is a typical state of matter.

Human and environmental health practitioners need to leave aside any dreams for ëblissfulí steady states in organisational dynamics and learn how to deal with critical states. It is the critical state where practical ability of the practitioner is tested, and not in the steady state of affairs. The latter simply does not exist in ever changing flow of life.

(3) Chaotic dynamics are ëruledí by strange attractors - emerging phenomena with whimsically strange forms (seen when mapped on the phase space - a mathematical space containing all possible states of dynamical systems). Anything off the strange attractor is ëfoldedítowards it, but anything on it is ëstretchedí in an unpredictable way - except that one thing is predictable: it always stays on the attractor.

When dealing with complexity of an individual or an organization or a society as a whole, we need to understand what kind of strange attractors are propelling the complex dynamics of this individual, organisation or society? What fields of social activity attract, inspire and concentrate the energy of people and what are the regions where this energy dissipates? Are there any hidden forces responsible for bringing forth specific organisational dynamics? If it happens that the present strange attractors are detrimental for human survival, how can the emergence of new attractors be catalyzed which are in favour of personal (or/and organisational) growth?

(4) Chaos is ubiquitous: a chaotic orbit can come arbitrarily close to any point in the phase space.

Since chaos can occur on all size scales of human activity, people could use instabilities in order to manipulate the motion of energy in the society on a very large scale. This becomes possible due to the butterfly effect: little changes bring forth significant results. The butterfly effect gives an incredible power to the hands, brains and hearts of practioners working in the field of human and environmental health.

When properly used, this power can produce essential improvements in the health of people and their environment. What a great responsibility for peopleís action in the age of Chaos and Complexity!

II. Second discovery: Reducing does not simplify: interaction is important and interaction means inseparability.

Chaos and Complexity entirely deniy the idea of analyzability of the world into separated parts. Parts simply do not exist; the Whole consists of Wholes. We can never understand how a specific stream of system dynamic ëworksí, without looking at it as a manifestation of the activity of the whole system. We can never grasp the integrity of human and environmental health, if we do not see it inherently imbedded in a higher order socio-ecological and economico-political structures.

Fractals, discovered by Mandelbrot, are similar patterns repeating themselves at higher orders of dimension. If you magnify a fine area of fractal structure, you get increased information in proportion to the new scale. Thus, the world not only looks different to the observers at different scales, it also measures differently. In every day language, this powerful mathematical insight means that the deeper your understanding of a complex picture, the more meaningful nuances you can notice in it.

In biology we can see that every cell of the organism bears the unique genetic code of the whole organism, otherwise it would not be able to interact with other cells. Conversly, it is also understood that it is because of the interactive nature of cells that cells have elaborated this code. The cells make the organism function, and at the same time the organism as a whole supports the functioning of each cell.

Another insight of biology which proves the unbroken wholeness of life relates to the aproposis - an essential property of all living entities: cells are geneticaly programmed to kill themselves, and it is only the constant support from their neighbours that keep them alive. No one cell is capable of living in isolation. No one human being either!

This unbroken wholeness in which, and through which, we exist is another manifestation of self-reflexivity, which was brillantly captured by Gödel in the proof of his celebrated Theorem: it is possible to make true statements within a particular system that cannot be proved by use of the elements and logic of that system. This is simply because the system under consideration is organically interconnected with some larger system, which by itself is dissolved into another greater than it, and so on.

III. Third discovery: Simple linear causality does not apply to Chaos and Complexity.

When the entire universe contributes to the existence of even a ësmallestí thing, how could we distinguish any specific cause-and-effect relationship in this universe?

Where everything relates to everything else in a tangled dynamic web of interdependent relationship, how can we speak about any linear cause-and-effect analysis? This analysis can work in general systems theory, where causality is always one way-directed: from the parts to the whole.

This discovery gives us a key for understanding the crucial difference between the "classics" of Systems Science and the "heresy" of the new Science of Chaos and Complexity.

According to the "classics", an effective description of a system can always be built from the descriptions of its separately analyzable parts. What is needed is to identify the forces which help keep the parts of that system in a balanced relationship, and to remove the perturbations pushing the system out of equilibrium. Although this approach never works in reality, the practitioners continue to use it in pursuit of illusory states of equilibrium of the system of human and environmental health. As such states never exist, they waste time and resources pursuing mirages.

According to the "heresy", separately analysed components can never create an adequate description of the whole. Positive and negative feedback loops are permanently driving the overall behaviour out of equilibrium, towards the Edge of Chaos - a critical zone between disorder and order, where emergence of new qualitative states takes place, and transformation of the system as a whole occurs.

Why is the Edge of Chaos such an important place?

Inside the chaotic region systems may be overwhelmed with change. In the ëfrození region of order they would not be able to sufficiently adapt. It is just on the frontier between these two regions - at the Edge of Chaos, where a delicate dynamic balance between chaos and order emerges, impregnated with seeds of innovative transformations.

While we cannot be masters of transformation at the Edge of Chaos, we are not slaves to it. We co-create it! This is an entirely new challenging way of perceiving the role of the human and environmental health practitioners in the age of Chaos and Complexity.

"In a recursive, complexly interwoven world, whatever one does propagates outward, returnes, recycles and comes back in a completely unpredictable form. We can never fully know to what results our action leads. We take action, the action can have a very potent shaping effect. Then we relax the drive to control and allow the process to unfold - the process learns, shapes and changes itself through all its inseparable components, not under the direction of one of them only. Together with overall changes in the process, we also change, almost unnoticeably, without any strain"... (Goerner, 1994)

IV. Fourth discovery: Complex dynamics give birth to forces of self-organisation: the self-organisational force seems to arise spontaneously from ëdisorderedí conditions, not driven by known physical laws.

How can entirely new structures emerge from the multitude of interactions within the complex systems? The concept of vorticity explains this stunning phenomenon. "When the vortex is swirling you could swear that there is a force somewhere. Where is it coming from? The answer is perhaps the most fundamental acknowledgement in all of Complexity: it comes from within the system. Although there seems to be an external force organizing the vortex, it is the masses in the vortex that is driving it" (O. Am, 1994).

The self-organising force of a vortex cannot emerge unless the participating streams - either masses or running water or turbullent air, or flows of irritating ideas or burning emotions, are:

(i) permanently in motion, that is, out of balance, and
(ii) interacting intensively with each other.

When these two conditions are satisfied, the sucking force at the centre of the vortex spontaneously appears. Maelstroms in troubled waters or tornados in fierce skies are mighty examples of how this force acts.

In the ancient Sanskrit books of yoga, seven major spinal chakras of human body are described - each chakra representing a swirling vortex of a vital life energy. Meditation practice and yoga exercises help to awake the dormant forces hidden in the vortices and thus are thought to improve human health.

Vortices of immense energy are hidden in human group dynamics. It is a noble challenge for the health and environmental practitioners to wake them to life. The ability to inspire and ignite imagination, warm hearts and awaken hopes and aspirations, radiate joy and stimulate new thoughts and vision - this is what brings forth self-organising forces out of complexity in organisations. If this is done with honesty and humility, good will and sincere desire to help and support, the vortices continue liberating peopleís potential for creativity and growth.

Fuzzy Logic: A practical Tool for Working with Social Complexity
Fuzzy Logic, proposed by Zadeh in 1965, is a method for understanding, quantifying and dealing with vague, ambiguous and uncertain qualitative ideas and judgements. Social aspects of Fuzzy Logic has been studied by Dimitrov (1977).

Fuzzy Logic is based on the following principles:

(i) principle of incompatibility: as complexity rises, precise statements lose meaning and meaningful statements lose precision;
(ii) principle of parallelism: a statement can be true and false up to some degree in parallel.
In classical logic statements can be either true (1) or false (0).

Fuzzy Logic tolerates a continuous spectrum of truth values between ëcompletely true (1) and ëcompletely false (0).

Human concepts, opinions, judgements and expectations change and evolve in the dynamics of real-life complexity. This complexity appears to us paradoxical and chaotic.

Social complexity is paradoxical, because it is a source of many contradictory and opposing forces that may act together, while at the same time, it is a product of these forces. For example, the forces produced through our physical and mental energy intensively act upon natural (and social) environments in different directions and up to different degrees, changing them permanently. At the same time natural (and social) forces intensively act upon us, again in different directions and up to different degrees, changing us permanently. These co-evolving interactions breathe paradoxes into our existence.

Social complexity is chaotic because the dynamics of real-life complexity is both unpredictable and sensitive to changes in the magnitude and location of these forces.

Fuzzy Logic helps in describing, analysing, understanding and eventually dealing with the paradoxical and chaotic nature of social complexity.

1. How does Fuzzy Logic help deal with social paradoxes?

Simply by tolerating opposites, by balancing them to such a degree that they cease to cancel each other out, and become complementary.

With Fuzzy Logic we can create an imprecise and easy to re-shape and modify framework in which the "either/or" approach to the contradictory concepts expressed in the paradox is replaced by a "both/and" relationship of their parallel acceptance. For example, a fuzzy framework created in management practice can meaningfully transform expressions like "collaboration OR competitiveness" into "collaboration AND competitiveness", "re-organisation OR stability" into "re-organisation AND stability".
2. How does Fuzzy Logic help deal with chaotic patterns of behaviour in social systems?

By generating fuzzy heuristic rules - rules emerging from human experience, which focus on the local, the decentred and the marginal in social behaviour.

Often what is excluded, what is not seen in the patterns, ëwhat is not thereí becomes important in constructing a list of fuzzy rules to describe and grasp ëwhat is thereí, what is in the core of the system, what makes the system chaotic.

We usually describe situations in particular, habitual ways, according to our, and our colleagues mindsets. However we can also recognise these as being limited, excluding some explanations; the latter become the what is not there that need to somehow be brought into consciousness and consideration.

A chaotic social reality paradoxically seems to unite people in a common desire to be different, unique and creative. Plural descriptions, open for change, based on peopleís personal experience and shared through dialogue, or derived by collaborative inquiry, are meaningful in such a reality - they help to disclose the tensions, not to resolve them, but to promote the examination of the contradictions and inconsistencies, and by the same token to reveal the conditions which could trigger the emergence of some kind of order (transitory though it usually is) in social systems. 

The principles of non-exclusion and non-isolation are fundamental to the use of fuzzy logic in any turbulent social environment.

Non-exclusion means that no options or alternatives, however improbable they seem to be for inclusion in future scenarios, should be excluded from consideration. They might turn out to be of crucial significance for the survival of society and its environment.

Non-isolation means that chaotic behaviour does not privilege one optimal solution. To isolate only one option, alternative or strategy bydescribing it as the best, the optimal, or the most efficient, from whatever point of view, is senseless; turbulent dynamics do not tolerate any pre-imposed isolation, however 'optimal' it may appear to the decision-maker.

The paradoxical and chaotic nature of social reality causes a great deal of uncertainty and vagueness in human decision-making. Under conditions of uncertainty and vagueness, when no ultimate answers or best solutions exist, the dialog and search for understanding and for consensus between people becomes crucial for the management of social complexity.

Fuzzy Logic opens the eyes of the practioners in the field of human and environmental health to the futility of black-and-white dreams of universality, 'unique solutions' and 'best answers'. With Fuzzy Logic practitioners learn not only how to be tolerant to different and contradictory viewpoints, but also how to look for and create conditions which facilitate the switch from contradictory value-differences to complementary ones.

And this is the way of wisdom.

1. Lorenz, E 1993 The Essence of Chaos Washington: University of Washington Press.
2. Mandelbrot, B 1982 The Fractal Geometry of Nature New York: Freeman and Co.
3. Waldrop, M 1992 Complexity London: Macmillan.
4. Stewart, I 1989 Does God Play Dice London: Basil Blackwell.
5. McNeil, D and Freiberg, P 1993 Fuzzy Logic Melbbourne: Bookman Press.
6. Am, O 1994 Back to Basics: Introduction to Systems Theory and Complexity. Personal Communication.
7. Dimitrov, V, Woog, R and Kuhn-White, L 1996. The Divergence Syndrome in Social Systems. In Complex Systems 96, eds. R. Stocker et all, Amsterdam: IOS Press.

(January 2001)

VLADIMIR DIMITROV is a researcher at the Centre for Systemic Development of the University of Western Sydney – Hawkesbury, Australia. 


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