Part 4. Towards a Theory of Complexity


Chapter 11. Stability and Chaos
Chapter 12. The Dynamics of Order and Disorder
Chapter 13. A Law of 
Entropic Cycling


Chapter 11

Stability and Chaos



He that will not apply new remedies must expect new evils, for Time is the greatest innovator

Bacon Essays, 1625

Stability is associated, conventionally, with some low energy state: open, interacting systems tend to dynamic stability with high energy operating at the edge of chaos. In Part 4, the search is on for a Theory of Complexity, a counterpart to the Theory of Everything. On what would such a Theory of Complexity be based? It must at a minimum consider:-

  • The dynamics of complex systems
  • Sources and generators of complexity
  • The stability of complex, open, interacting systems
  • Models of Complex Behaviour
  • Laws of Complex Behaviour, offering: -
    • - explanations of phenomena
    • - predictions from the Laws

Given all of these, we might have a Theory of Complexity which would be both useful and capable of challenge.

Dynamic Systems Model

Examination of open interacting systems reveals that activity levels increase with the energy available to the systems. Energy entering a set of interacting systems: -

  • drives activity, or...
  • drives speciation, or...
  • both

Activity and/or speciation increases or decreases until the rate of energy absorbed in the internal processes equals that entering the system. Put at its simplest, more energy results in more activity and/or more speciation.

What evidence is there to substantiate such a claim? For physical and chemical systems there is unlikely to be much dispute, since the behaviour of such systems is so readily observable. Heat a gas and the molecules move faster; temperature may be defined by the kinetic behaviour of a gas. Sugar dissolves faster in hot tea rather than cold water. Chemical reactions are accelerated by the addition of heat energy, from the washing of clothes and dishes, to the speeding of catalysis.

For ecologies, the evidence is similarly simple but persuasive. Growth and decay in tropical rain forests operates at up to five times faster than in moderate climates. This evidence is particularly interesting since, unlike the simpler physical and chemical examples, ecologies exhibit such behaviour while encompassing very high levels of complexity, with many interacting systems in many interwoven hierarchies.

Energy entering an open system promotes: growth; activity; interaction, including (where appropriate) reproduction and speciation; decay; and recycling. There is a marked tendency for complex systems to cycle continually.

Sources and Generators of Complexity

In social and economic systems, energy may be generated by powerful ideologies or by the pursuit of wealth, status, happiness, promotion, etc. The energy within such systems may be seen as the potential energy of wealth, visions, ideals, etc.

Speciation, usually a biological notion, appears around us in many other systems: -

  • product variants, e.g. car models, building designs, washing machines, food mixers, clothing fashions, perfumes and jewellery, washing powders, etc.
  • infrastructure densities, e.g. telephones, sewers, roads, railways, computer networks...

We tend not to think about speciation of material things, but their variety clearly increases with wealth and freedom or openness...

Natural and Man-made Environments

Fractal 9. "Chicken Legs"

We are used to thinking in terms of three physical dimensions, with time being a fourth. It is simple enough to increase these numbers, and to see complex things rather more simply in the process. Conversely, things with many dimensions, N-dimensional entities, can be reduced to the familiar two or three to advantage also.

Suppose we allow dimensions to include rather more than up, sideways and forwards. Suppose we allow: -

  • temperature, temperature range,
  • humidity, humidity range,
  • pressure, pressure range,
  • foods,
  • toxins,
  • predators,
  • hosts,
  • co-operators, etc., etc.

We can put these together as dimensions. So, a complex natural environment may contain a space, for a time, which has a high but stable temperature, accompanied by humidity varying between 90-100%, at high altitude/low pressure, with a range of lichen and mosses, no toxins and no predators.

We have described a micro-environment which might offer a "niche" to a particular species which evolved to make use of it - after all, it appeared to be safe. N-dimensional spaces offer many such niches - see Fractal 9; flora and fauna which evolve, perhaps chaotically, will eventually speciate to produce something which will shelter and flourish in particular niches. It is as though the availability of a niche determines the survival of species.

How, then, do niches fill? They fill to an extent, and at a rate, determined by rates of activity and rates of speciation. The greater the "free" energy within the environment, the faster the N-dimensional environment will be filled. But it is not as simple as that; consider both Fractals 9 and 10, which try to illustrate that which cannot be drawn - N-dimensional space. Filling a multi-dimensional space could be tackled by a number of quite different strategies. We have mentioned chaotic meanderings but, in a man-made approach, we could consider linear control, with planned developments of people and activities working in co-ordination moving in a series of parallel straight lines, meeting an obstruction, re-planning and setting off in a new direction... Which strategy will find and fill all the niches in the available space: most fully; fastest?

Now, we were not really considering natural environments in the last paragraph; instead we were considering business ecologies, economies and perhaps even crime. In many respects each of these expands within the available space to occupy the available niches which allow them to flourish. Questions of the fastest strategy for occupying the most niches is far from academic - it is vitally important to social and economic development and underpins differing political ideologies.

Fractal 10. N-dimensions?

Nature’s pattern of generating activity and variety to match and absorb available energy is proposed, then, as general and extendible to all systems, including those of politics, economics and business. Political environment dimensions include: social climate, power structures, concentrations of wealth, cultural differences, the rates of change of these, and many more. Economic environment dimensions include: concentrations of wealth, wealth generators, wealth flow, rates of change of these, and many more. Business environment dimensions include: money supply, market demand, skills availability, competitive threat, rates of change of these, and many more.

As with ecologies, it is the very complexity of these multi-dimensional environments that creates niches, which will be filled either by design, or by chaotic speciation. As with ecologies, it is the niches which seem to determine which species survive and flourish and for how long - substantially different from the political ideology of free market competition. To determine which of these approaches might be the more successful, consider open system stability.


Animals and plants can stabilize in mass without feedback. Organizations can stabilize without feedback. Not that feedback need be absent. For example, flora and fauna exhibit satiation, which tends to prevent further immediate inflow: humans do feel full-up (feedback) as they eat, stopping further eating and stimulating subsequent excretion some twenty minutes after the onset of satiation. But such feedback is not essential to stabilization; people who continually nibble at food may never be satiated, but do not grow forever; they stabilize at a mass which reflects a total intake/outflow balance.

Open systems stabilize at high energy levels, not low. The human stabilizes at high mass/potential energy. Interacting systems may be in a state of constant flux; they appear to be unstable in the short-term, but stable in the longer-term. Stability is particularly difficult to define universally, but a new definition of stability is essential to address open systems. The following definition is offered: -

A set of interacting systems, itself constituting an open system, may be said to be stable when, over a period of interest, its net configuration entropy tends to a constant value.

As real-world systems are open, this definition must apply to any real world system, or system of systems.

In open systems, there seem to be several routes to stability: -

  • Complementary Sets - groups of systems with complementary interflows providing some or all of their mutual needs, reducing both local entropy and the need to re-arrange - this reduced need to change is a move towards stability
  • Preferred Patterns - positive feedback resulting in multiple points of stability. Interacting systems with positive feedback tend to lock into one stable point, which can prove very resistant to subsequent change i.e. stable.
  • Dominance in natural and social systems. Ancient Egypt was a social system dominated by a transcendental culture; it proved very stable over a long period.
  • Linear stability - systems interact linearly within an open set. This may lead to relative ease of design and management, but is highly likely to result in counter-intuitive responses as the linear systems interact with other, generally non-linear systems, e.g. Planned Economy interacting with the real, chaotic world of commerce and agriculture.
  • Catastrophic stability - systems interact non-linearly, between extremes. Such systems can be long-term very stable, e.g. two party politics, function/project switching
  • Chaotic stability - members of set behave chaotically, i.e. uncoordinated. Can be very stable, but indeterminate, e.g. Free Market Economy

Stability in Open Systems

Stability in open systems is really quite unlike stability in closed systems. As the simplest example, consider Figure 64, which shows a bath with taps open and the plug out. As the bath fills, the head of water and hence the outlet pressure increase until a point may be reached at which the level of water in the bath stabilizes. There is no feedback from outflow to inflow. Like other open systems, stability occurs at high energy, rather than low, represented in this case by the potential energy of the water in the bath.

Figure 64. A bath with a constant rate of water flowing in will, provided it is deep enough, eventually reach a stable level because the rate of outflow increases as the bath fills. Eventually, outflow rate must equal inflow rate - stability

Open systems can oscillate without feedback

Figure 65. Oscillation in an open system without feedback, and with only one energy store

The bath analogy is continued in Figure 65, where the model represents a constant inflow to (initially) empty Reservoir or bath. As before, the outflow is proportional to Reservoir contents, but there is a fixed time delay between the level in the reservoir changing and the consequent outflow rate. This delay might equate to a long drain pipe. The result, as the graph shows, is oscillation, without feedback, and with only one energy storage component - the reservoir itself. We are not unfamiliar with this phenomenon: sink outflows gurgle; candle flames gutter.

In these examples, the reservoir and the flame, the resultant state is dynamically stable in the sense that the mean value tends to a constant value over time. Other well-known systems of this general nature include: -

  • Deciduous trees - leaf-fall is proportional to the number of leaves in growth, but is delayed by the seasonal changes
  • Delivery systems - deliveries are proportional to the number of products in manufacture, but delayed by the need to accumulate a viable load
  • Government budgets - constraints or de-restrictions are proportional to the number of problems caused, but delayed by the annual budgeting cycle and by the time taken to perceive the problems

In each case, the open system may oscillate simply due to the situational structure.

Bounded Stable Chaos And Poincaré Maps

This book is not about chaos as a subject, there are many books available (see Lauwerier, 1991 for an interesting introduction to fractals and chaos). This book is illustrated with fractals, signifying the generation of (sometimes beautiful) complex systems from very simple sources. Some of the features of chaotic systems have already been introduced in Chapter 2, and Fractal 11 illustrates some further chaotic principles.

Fractal 11 shows a progressive development of a Poincaré Map reading from top left. The map is created by simple interlinked, non-linear equations which create orbits in a 3-D space. As an orbit passes through the plane of the Map, a single dot is recorded. The orbit continues on its path, returning through the plane of the map repeatedly, building the pattern of dots. Watching the build-up, it is difficult to predict where the next dot is going to appear, but - as the figure illustrates - it soon becomes evident where no dots will appear, and where dots are more likely to appear.

The endless fascination of chaotic phenomena is in the complexity of pattern that can be generated by very simple equations. The beauty and elegance of the Poincaré Map is in its ability to show at a glance the essence of that complexity, its clearly-defined boundaries, and the paradox between its short term unpredictability and its long-term predictability. The Poincaré Map also reduces perceived complexity by reducing the number of dimensions in its presentation.

Fractal 11. Progressive Development of a Poincaré Map - de Jong Chaos

To bring the Poincaré Map and chaos down to earth consider football. Imagine a football pitch, with an electronic detector system which detects the football as it passes through a vertical plane, parallel to the goal posts. The plane may be set anywhere between the goals, but consider it set, say, 15 metres from one of the goals. Two teams start to play, and the detector starts to build a Poincaré map, recording each time that the ball goes through the plane. What would the developing pattern look like?

In general terms, it might look like Fractal 11, or at least the top halves of each of the panels, since the ball cannot go below ground level. I suspect that it would display a sort of "footprint" of the interaction between the two teams. Suppose that the two teams played a second, and a third time. Would the same pattern emerge? Suppose, instead, that we followed the fortune of one team, as it played a number of different opponents. Would the pattern of play of our team imprint itself as a signature, modified but not obliterated by each opponent? I suspect so, because the Poincaré map would record the essence of the team’s style of play, whether they were an attacking team, whether forwards went down the wings or through the centre, whether they used their heads or their feet, and so on.

Suppose, now, that we decided to write a simulation program, to emulate and predict a game of football, from the moment of kick-off to the end. What prospects would there be of the simulation matching reality? For instance, would the program predict where the ball and all the players would be in the 15th second of the 73rd minute? Of course not, such a program could never work. At each kick, on each bounce, with each turn, for every sprint, slide and tackle, there is an uncertainty, an imprecision that defies any notion of simple prediction of effect from cause. Football is short-term unstable, long-term stable, short-term unpredictable, long-term predictable - at least in terms of trends. Will a first division team beat a fourth division team? We may not be able to predict the kick-by-kick play, but the outcome is less uncertain.

What is true for football is true for much of life. Instead of football teams, substitute competing businesses, competing political parties, competing economies, rival families, opposing armies...the list is endless. While each of these may appear to be highly complex, and while their blow-by-blow activities may seem unpredictable, there is a pattern, a gestalt, an essence which determines the trend. It is this essence which the social genotype typifies, and to which Belief Systems and shared visions give energy and direction. These underlying patterns may be quite simple, but, as the Poincaré Map shows, we have to develop the knack of standing back in order to see them.

It is more in our natures, it seems, to concentrate on the minutiae, to disassemble rather than to synthesize. In so doing, we may create well-ordered systems only to be disconcerted by their lack of success. Consider football once more; a well-ordered system for playing football is zone defence, with every player allocated a zone; it is often the way beginners are taught to play. Top professional teams tend to be much more fluid in their play, with individuals roaming freely over the pitch, aware of the whole game, probing for openings, backing up team mates, and generally exhibiting superior ball skills. This type of play will almost always overcome zone defence, which is cumbersome, and where loss of a player leaves a hole. Zone defence interferes with synergy and flow between the players. For business or warfare, the parallels apply; structure and order have their place, but they can inhibit synergy, flexibility and flow, and may cause fatal predictability.

If we think about players as open, interacting systems, and if their behaviour is chaotic yet consistent, then the question of this longer term stability arises. How can open systems, often without feedback, be stable in a chaotic environment?

Chaotic Systems at Work

As Fractal 10 showed, chaotic systems may be well bounded and often predictable in terms of what they will not do. Many chaotic systems are long-term stable, short-term unstable, including: the weather, the stock market, war & conflict, competitive sport.

Consider the football team again, if only because it is such a well-recognized analogy. There may be chaotic activity, but the whole team is working to a strategy, aiming for a goal. Seen by itself, the path of an individual player presents a meandering path, given meaning only when whole game is seen. Consider engineers in a manufacturing organization looking for new tasks, proactively seeking problems to solve; there is a direct analogy to the football team.

Consider marketing for a new business; is the marketing forecast intended to predict an unpredictable future? Some companies think so, and measure marketing success by accuracy of prediction, not be the level of business achieved.

Chaotic Stability

Chaotic systems present difficulties in predicting the next event, but are not random. Chaos arises in non-linear systems where there may be many concurrent, but uncorrelated events, and where there are chains of events such that each outcome is related to some previous events or causes.

A free-market economy operates on the basis of a wide variety of companies, each unco-ordinated with respect to the others, sharing the same profit goal; it is this common goal that offers chaotic stability. Nature, similarly, evokes a multitude of concurrent, uncorrelated activities to maintain the coherence of populations and to gradually evolve predator / prey performance. Chaos, paradoxically perhaps, seems to contain the seeds of stability. It is ironic, then, that individuals within organizations seek to eliminate the chaos, to so regulate and control activities into linear, "well-behaved" processes that any variability is driven out. It seems more than likely, even inevitable, that their efforts - if successful - must inevitably lead to their eventual downfall! Happily, our organizations may be so transient that they, too, rise and fall chaotically to create a dynamic stability - observing the effects of chaos requires that we stipulate a time frame!

Chaotic Stability Simulation

Sampling theory shows that the proportional variability between samples goes down as the size of the sample increases. Pollsters trying to predict the outcome of an election are aware - painfully aware, judging by their lack of success in recent years - that small samples can give erroneous results. This idea of sampling is reflected in the following models.

Figure 66 presents three very simple models of a system, marked Level 1, Level 2 and Level 3 respectively. Each model has the same mean inflow and outflow, except that Level 1 has one source and sink, Level 2 has five sources and sinks and Level 3 has ten sources and sinks: -

  • Each inflow and outflow is independently random
  • Mean inflow is the same in each model
  • Mean outflow is the same in each model

Provided that the inflows are mutually independent, and the outflows are also mutually independent, it might be expected that peaks and troughs in flow would tend to iron out rather more when there are more sources and sinks than when there are less.

Figure 66. Three representations of unco-ordinated inflow/outflow systems


Figure 67. Simulation results from the models of Figure 66

Figure 67 confirms that expectation. Level 1, at the top, is much more variable than Level 2, with five inflows and five outflows; Level 2 is, in turn, more variable than Level 3 with ten inflows and ten outflows.

As expected, variability decreases with the number of random sources and sinks - provided they are mutually independent. Clearly this is relevant to a free market. A supermarket may keep its shelves stocked by buying the same basic product from a number of different suppliers and, provided they are independent, the supermarket stands little chance of running out. Similarly, we can pop into any supermarket on the off chance, and can be reasonably sure of picking up what we want, if not the precise brand. So what? So this pattern of chaotic supply and use may give a clue to evolution.

Chaos - Essential for Life and Evolution?

Variability decreases with the number of random sources and sinks; more sources and sinks enable systems to reduce stock levels without fear of shortage - chaotic systems can therefore be more efficient and are more likely to survive lean times. In Darwinian terms, there is survival advantage in chaotic efficiency. Life self-sustains through a web of interactions between a variety of species - variety being crucial. So: -

  • chaotic interflows are more efficient and survivable
  • chaotic systems sustain variety, by avoiding domination
  • variety is essential to life

The inevitable question arises: was/is chaos essential for life to evolve? It seems highly likely. If we accept that chaos may have been essential for Darwinian evolution, then is chaos essential for social evolution, too? That is certainly not the general view, but it is an interesting proposition.

Variety, Synthesis and Decomposition

Figure 68 shows a highly-simplified representation of a socio-economic systems set. The set is complementary. The representation of Figure 68 enables us to see chains of interactions. For example, Farming Industries provide food to Society, which offers human resources and dated skills to various Service Industries. In turn, these Service Industries provide power, distribute goods, transport freight and people, and provide communications to Society, which also provides human resources to work in the Farming Industries. That description moved through a closed double helix, a closed chain of interactions in Figure 68.

In the figure there are: one 5-loop, five 4-loops, ten 3-loops and ten 2-loops. Moreover, each "system" (e.g. Service Industries) comprises many concurrent, possibly unco-ordinated and competing elements. The potential for social genotype development is enormous. Stability within such a complex structure - and the figure is highly simplified - could be controlled/planned or chaotic. We have already met the simulation of such a system - see Figures 17 and 18. Simulation shows that the long term effects of substituting, say, imported raw material or food supply in place of a home grown commodity impacts all other systems, each gradually moving to a new (uncertain) state affording overall stability, but with every part of the system operating at a different level. It takes a long time for the effects to permeate through the environment, but permeate they do - according to the model.

Figure 68. Notional chart showing complementary social and industrial systems interchanging variety, synthesizing and decomposing (as in food chain) to create a dynamically stable economy/society, resistant to change and damage, but requiring continual energy-pumping to enable processes. Outputs from the leading-diagonal systems are in their respective rows, inputs in their respective columns, and variety goes into the page. (Compare with Figure 17.)

Free Market Competition

In a free market, a new point of stability appears, paradoxically, to be both guaranteed and yet unknowable. However, there are conditions for a free market to exist, chief amongst which are that, like an ecology, it should have a richness both of variety and of interaction.

Figure 69. Competition in a restricted environment. The inner loop is negative feedback, progressively reducing competition until, eventually, there is a monopoly.

Genuine competition invokes losers as well as winners. In Nature's ecology, losers often die, but that would be far too expensive a fate for economies. Instead losers must be able to lose yet survive - at least several times.

Suppose, however, there is a small market, with many competitors. There will, inevitably be more losers than winners. See Figure 69. Over time, the variety of organizations able to compete inevitably reduces, losers switch/go out of business. The result is inevitable - monopoly through survival. The only question is: how long does it take? Such questions never get asked, let alone answered, when political ideologies and beliefs rule.

Fractal 12. Ikeda


Chapter 12

The Dynamics of Order and Disorder



It is the nature of an hypothesis, when once a man has conceived it, that it assimilates everything to itself as proper nourishment; and from that first moment of your begetting it, it generally grows the stronger by everything to see, hear, read or understand. This is of great use.

Laurence Sterne 1713-1768

Towards a Theory of Complexity

We have seen that complexity is made up of: variety, connectedness of that variety, and disorderliness of both. We have also seen that symmetry can engender minimum energy of exchange in complex systems, offering survival advantage. It seems, too, that open, interacting systems naturally evolve dynamic stability of synthesis, decomposition - at the "edge of chaos".

Social, organizational and transcendental systems evolve genotypes analogous to Nature’s. The social genotype hypothesis proposes how and why organizational/social systems resist change, adapt and evolve only slowly. The social genotype may even be seen more as an extension of the biological genotype than merely an analogy.

Behind these phenomena and perceptions we see, not control and regulation, but complementation and connected variety as the bases for open, interacting systems stability. It is also evident that open, interacting systems stability breaks down when connected variety reduces in a changing environment, owing to inability to adapt to changing environment.

Lastly, it seems that chaotic systems are potentially more robust than those we seek to control; this paradox is at its most apparent when comparing planned and free-market economies. No matter how careful the planning of the former, stability breaks down with time. On the other hand, the free market economy, while seemingly not breaking down, is unpredictable and may adopt temporary states that are less than desirable. Ecologies respond not dissimilarly.

These observations and perceptions are so consistent and widespread, apparently covering systems of all types and sizes, that the possibility of some straightforward underlying theory must be considered. In seeking any theory, it is useful to find invariances. One such invariant is that systems come into being, exist for a time, then decay or collapse, to be replaced by other systems. Evidently, there are opposing influences, some promoting stability, others promoting decay, the whole promoting "life-cycling" - the continual formation and breakdown of open, interacting systems.

Open Systems Interactions

We can make some general observations, based on previous chapters: -

  • Energy creates disorder. If we heat a substance, its molecules vibrate, there may be a change of state. If energy is injected into an argument, disorder may result. If solar energy is radiated into an ecology, speciation occurs, activity levels increase, movement increases
  • Variety interacts to form systems. Sometimes the interaction is uneventful, sometimes inimical, but sometimes the parts find mutual benefit. On those occasions, separation would be unnecessary or disadvantageous, and the varieties may form a system
  • Systems reduce entropy. Because of the complementary nature of the parts from which a system must be formed, both the parts and hence their intra-connections form a pattern. That pattern represents reduced (configuration) entropy, a localized ordering against a disordered backdrop
  • Systems break apart sooner or later, increasing entropy back towards the state before the system formed
  • Previous bullet-points above present a concept of continual change over time in which: -
    • - the mean level of disorder is increased by energy
    • - the rate of change from order to disorder to order is increased by energy

Energy, it seems, not only creates disorder. For open interacting systems it also creates order from that disorder.

Figure 70. Open Interacting System Analogy. Surface of a pool with submerged pump causing water to well up. The surface shows ephemeral whirlpools and eddies, each stable for a time, but each deforming, the water molecules becoming absorbed into other "systems". Each system is either forming from disorder, maintaining order, or becoming less ordered - hence each "observes" internal order, external disorder - "edge of chaos". The whole behaviour is pumped by the unseen energy source.

A useful analogy to understand this view of open interacting system life-cycling is given in Figure 70. It shows the surface of a pool which contains a submerged water-pump, causing the water to well up to the surface. The resulting surface is turbulent, with swirls and vortices forming and reforming, some lasting, some ephemeral, some large, some small, the whole presenting self-similar patterns, driven by the energy from the pump. Were the pump to stop, the patterns on the surface of the water would die away. Were the pump to increase its flow, the surface would become more turbulent, more systems would form and reform, and at a higher rate. At a still higher level of energy, different types of pattern would occur, with bubbles, spouts and fountains.

So it seems to be with all open interacting systems. An ecology employs energy from the sun, which causes growth, activity, formation of floral and faunal systems, decay and reformation. While the solar energy lasts, the life-cycling continues. Where solar energy is greater, the number of species and their rates of growth and decay increases - typically, rates of growth and decay are five times greater in a tropical rain forest than in a tundra.

Romme and Despain (1989) investigated recurrent fires in Yellowstone National Park. There were relatively few occurrence of major fires, although minor fires, initiated by dry weather and natural or man-made sources, occurred frequently. Between the early 1700s to the summer of 1988, there were major fires in 1690-1709, 1730-1749, 1850-1869 and 1988. Why the unexpected pattern?

After each fire, the ground was scorched. A few plants were adapted to survive fires, and their deep-lying structures grew to form a new, verdant growth of initially-soft plants. Birds, bats, animals and the wind brought seeds into the area. After a time, softwood trees started to grow, shading the undergrowth and taking some of the sun's energy for themselves. As a result, the pattern of forest floor plants changed, with different species emerging that were more adapted to these conditions. As softwood trees eventually died, they rotted down in the undergrowth under the action of decomposing bacteria, providing a mulch for new growth.

Hardwood trees emerged, with their slower growth. Hardwoods grew taller, creating a canopy with which to catch most of the sun's energy, shading the forest floor and drawing up moisture and nutrients. The forest floor plants died away, deprived of light, moisture and nutrient. Eventually, the dominating hardwoods died too, but when they crashed down, they did not create a moist mulch; instead they created a dry tinder which could support a major fire. One lightning strike in the right place would set the whole cycle in motion again.

The Yellowstone Park cycle presents a striking example of life-cycling. It is all there: speciation, variety, stability, emergence of dominance, suppression of variety by dominance, tendency to moribund, monotonal system, collapse, speciation... Note how dominance suppressed variety by starving lesser species of sustenance, but in so doing brought about its own downfall.

Civilizations behave similarly. The Ancient Egyptian civilization grew rapidly using energy from the sun and fertility from the Nile. With such exceptional and dependable energy, the civilization produced a profusion of open interacting systems including the cultural and transcendental systems introduced earlier. Dynasties came and went. Egypt conquered many lands and took tribute to maintain the flow of energy into their civilization. Eventually, however, the civilization decayed as other lands grew stronger, and as its internal political and economic systems became moribund.

Businesses behave similarly, rising and waning with greater frequency. IBM, like Ancient Egypt, grew particularly large, and dominated in part by virtue of being first on the scene. IBM was slow to adapt to the advent of smaller, networked desktop computing systems, clinging instead to the larger mainframe machines which had made it great. Recent tribulations within IBM mirror those in Ancient Egypt and in Yellowstone Park.

Nature and the Second Law

One of the central tenets of classical physics is the Second Law of Thermodynamics. It can be stated in a variety of ways, including: -

"The processes most likely to occur in an isolated system are those in which entropy either increases or remains constant"

The Second Law has been used to predict the gradual disordering and eventual cooling of Universe, and it is evident for many physical phenomena.

A key phrase in the Law is " an isolated system...". But the natural and social worlds are not isolated. So, does the Second Law apply, and if so, how? This appears to be a substantial issue. While mixing milk and tea increases entropy, what about the synthesis of tea and milk production? These include: the synthesis of grass and tea-bushes; milk formation in cows; bottling, packing and distribution. All of these incur increasing order before the milk mixes with the tea. Looked at overall, has entropy increased or decreased?

Further, life has gradually evolved on Earth, and is still evolving today. It is difficult not to see this slow, inexorable evolution as increasing order. It is reasonable to conclude as follows: -

  • Nature does not challenge the Second Law of Thermodynamics, because Nature's evolution does not satisfy the condition of "an isolated system"
  • The Second Law holds little relevance for open, interacting systems which include all those around us. (If a system were truly isolated, how would we even know of its existence?)

We need a law to complement the Second Law of Thermodynamics that does apply to open interacting systems - the Law of Complexity. In seeking for such a Law, or Laws, consider first the Unified Systems Hypothesis.

The Unified Systems Hypothesis (HUSH)

As part of my research, I have developed a so-called Unified Systems Hypothesis USH) over several years. My students have taken to calling it Hitchins Unified Systems Hypothesis (HUSH) and, since that title results in a nicer acronym, please forgive me if I follow their lead. HUSH is more fully explained in Putting Systems To Work, (Hitchins, 1992), and in the journal Systems Practice (December 1993). A brief résumé of the principles follows.

There are seven principles which together comprise the Unified System Hypothesis. Individually, they may seem simple, even axiomatic. Together they may seem rather less self-evident. First the principles.

The Principle of System Reactions

The Principle of System Reactions is stated as follows: -

If a set of interacting systems is at equilibrium and, either a new system is introduced to the set, or one of the systems or interconnections undergoes change then, in so far as they are able, the other systems will rearrange themselves so as to oppose the change

An equivalent statement proposes that, after the perturbation, the set of systems moves towards some new equilibrium. The Principle, based on Le Chatelier's Principle, says nothing about the manner of that movement; it may be linear, non-linear, periodic, catastrophic or chaotic, but each response will eventually reach equilibrium. The contention of HUSH is that the principle applies equally to interactions between economic, political, ecological, biological, stellar, particle or any other aggregations which satisfy the definition, system.


The Principle of System Cohesion

The Principle of System Cohesion is stated as follows: -

Within a stable, interacting, system set, the net cohesive and dispersive influences are in balance

For physical systems such as a star with its planets or the quarks in a neutron, such a statement might seem axiomatic, or perhaps another way of looking at Newton's third law. The proposition, however, is that the principle applies to all interacting systems, social, economic, political, etc.


The Principle of System Adaptation

The Principle of System Adaptation is stated as follows: -

For continued system cohesion, the mean rate of system adaptation must equal or exceed the mean rate of change of environment

The principle is evident within biological systems, and is under-pinned by Darwin's theories. It seems likely that the sudden extinction of the dinosaurs, and of some early S. American cultures, are spectacular examples. The proposition is that it applies to all interacting, systems including man-made systems.


The Principle of Connected Variety

The Principle of Connected Variety is concerned with stability of interacting systems: -

Interacting systems stability increases with variety, and with the degree of connectivity of that variety within the environment

At first appearance, this Principle may seem obvious. Certainly, if we look at natural habitats, we see stability dependent upon a web of interactions between plants and animals, with food chains, survival, reproduction and natural selection creating a complex continuum. If we look at the human social world, we see similarly-complex webs of social interchange, to the extent that governments sometimes intervene in business to unravel the web.

In the UK, as already noted, there is a move to introduce "trust status" for hospitals and schools, by which is meant granting them relative independence so that they are freed from the web of bureaucracy surrounding state schools and hospitals. Once freed, the argument goes, these schools and hospitals will be free to change and hence to become more competitive. It is also true, of course that, once freed, schools and hospitals are free to make mistakes - the web of connected variety can be looked upon as a safety-net, too.

The Principle also invokes notions, expressed earlier, of minimum, useful, limited and excess variety. Not all variety need be connected within a set. Unless a minimum degree of connection is made, the set will not be viable. Beyond that, additional connections may be useful, for instance in enabling the system to tolerate some level of damage. Excess variety might include that which makes a set of interacting systems so stable as to resist desired change.

The Principle of Limited Variety

The Principle of Limited Variety is stated as follows:

Variety in interacting systems is limited by the available space and the degree of differentiation (degrees of freedom)

The principle is axiomatic once "space" and "degree of differentiation" have been established. This Principle is important because it provides an essential counterbalance to other Principles. The Principle of Connected Variety, for example, might suggest the notion that stability could be increased without limit simply by increasing (and connecting) variety. The Principle of Limited Variety shows that infinite stability cannot occur.

Space, as referred to in the Principle, has to be considered in context. For example, the space within a rain forest for species is increased by the amount of energy entering the forest, which increases the potential niches. The variety of species observed within that space is determined by our perception that one specimen is significantly different from another.

The Principle of Preferred Patterns

As the weave of interactions between systems becomes more complex, it is increasingly likely that feedback loops will arise, some perhaps acting through many successive systems and exchanges. The prospect increases of non-linear interacting system behaviour. The occurrence of positive feedback loops is to be expected, if only because of resulting delays, and leads to the Principle of Preferred Patterns:

The probability that interacting systems will adopt locally-stable configurations increases both with the variety of systems and with their connectivity.

Positive feedback has, in the past, been considered as the basis of instability. It can be shown, however, that positive feedback in a network of largely negative feedback relationships results in the adoption by the systems concerned of one of a number of possible, highly-stable states.

The Principle of Cyclic Progression

The last of the HUSH principles addresses the phenomenon which we all recognize, that systems do not last for ever. (The Second Law of Thermodynamics concurs, of course, but HUSH goes further.)

Such observations support the Principle of Cyclic Progression: -

Interconnected systems driven by an external energy source will tend to a cyclic progression in which system variety is generated, dominance emerges, suppresses the variety, the dominant mode decays or collapses, and survivors emerge to regenerate variety.

The weight of evidence suggests that there may indeed be a repeating pattern in systems where variety, the mediator of stability, is suppressed by dominance, which in turn leads to vulnerability through inability to change.


The seven HUSH Principles may be brought together into a single explanation of the life cycle of sets of open interacting systems. Figure 71 illustrates the HUSH Principles taken together as a set. To follow the Map, start at Energy, upper right, and follow the map around, generally moving clockwise. Energy drives the cycle; the more energy, the more variety and the more mobility within the variety. With more energy comes a greater probability of interaction between the varieties, some of which will result in complementary interaction, some in dispersive influences. Complementary interaction reduces the energy consumed by the complementary systems in question, since they have some of their sources and some of their sinks in the vicinity. Further interaction sees more complementary systems attaching themselves, or inimical connections disrupting those which had connected.

Figure 71. The HUSH Map, showing how the various HUSH Principles combine to illustrate the formation, life cycle and eventual degradation of sets of open, interacting systems

Further systems may join, if complementary. Their interchanges with the other varieties within the set may be continuous or stochastic. Sets of complementary, open, interacting systems may form, and may create a local "web" of stability surrounded by a sea of temporarily unconnected systems. One or two of the systems may influence the set, perhaps by transforming a key variety, or perhaps by transforming more inflows to outflows than others in the developing set; this is the onset of dominance. Dominance is not, of itself, deleterious to the set, indeed dominance can make the set behave in a stable manner, the dominant member influencing behaviour of the whole and, as it were, pointing all their transformations toward a particular direction. But dominance can suppress variety, just as a dominant tree can shut out the light, and drain all the sustenance from ground level plants, preventing their growth. Once that happens, the behaviour of the set tends towards that of the dominant system, the sets becomes "monotonal" and may not be able to adapt to change. This is the onset of degradation and collapse, after which the cycle starts again.

Working around Figure 71, it is possible to see each of the seven HUSH Principles. Together, they offer explanations consistent of complex system life cycle behaviour. From Figure 71: -

  • More energy creates faster environmental change, more adaptation, more reaction and interaction
  • More energy creates more speciation, raising the limits to variety
  • More reaction and interaction, together with more variety, increases the probability of complementary systems existing and coming together
  • Complementary systems provide a basis for stability through connected variety
  • Stability is maintained by continual resupply of connected variety (as the human body remains stable only if fed with appropriate food and drink) which in turn depends on continued energy supply
  • System cohesion develops as a balance between the connected variety, promoting stability, and unconnected varieties which continually influence dispersion of the cohesive set
  • System cohesion maintains its form through either chaotic/ catastrophic/linear/stochastic interaction, or through the development of preferred patterns leading also to dominance.
  • Dominant systems tend to absorb/monopolize energy, suppressing variety within the set either inadvertently or by design
  • Cohesive Systems, but with suppressed variety, remain stable until an environmental change or a dispersive influence requires them to adapt. If they have become monotonal, able to behave in one way only due to lack of variety, they degrade or may collapse
  • The degrading or collapsing set provides the variety of its parts to start the cycle again

Once this cycle, and particularly the reasons for its transition from state to state, are understood, interesting insights emerge. The state changes are more easily seen in Figure 72. From the figure, it is evident that the way to create and maintain cohesive interacting systems is to maintain the level of connected variety while at the same time combating any dispersive influences. The figure also shows that seemingly stable system sets may in fact be moribund, waiting only for a change in the environment to reveal their susceptibility. Typical of such systems are organizations that, to survive a recession, have divested themselves of useful variety in pursuit of so-called efficiency. The result of their efforts is the ability of the organization to do only one thing. Should that one thing not suit the re-awakening market at the end of recession, major change in the organization is inevitable. The figure does not suggest the existence of a simple, direct return from Moribund to Cohesive, it being necessary to traverse clockwise through the upper two states. For the super-efficient, impoverished variety organization, this means either total demise or at least major re-organization.

Suppose instead that the objective is to damage a Cohesive Interacting System, to render it unstable. Figures 71 and 72 suggest that there are, again, two complementary methods. First, eliminating variety will render the system set moribund; this may be sufficient in a dynamically-changing environment, but may be too slow otherwise. Second, there may be a need to introduce a dispersive influence. As we have seen from the Social Genotype, that dispersive influence could well be brought about by an assault on the culture or belief system as practised by opposing armies in war, using propaganda and false intelligence. A new, vigorous belief system would have a similar effect.

Figure 72. HUSH State-Transition Diagram, showing four states in the formation and dissolution of Open Interacting Systems, and the causes of switching between those states.

If the intent is to develop or maintain a long-lasting, extremely robust system, then a different tack may be necessary. The route to domination is unsuitable - that is the path to (possibly) effective, but short-lived stability. Instead, the path to follow is shown on Figure 71 as going direct from Tendency to Stability to Systems' Cohesion. There are several options along this route, but they share one feature - no domination. Probably the soundest route is towards chaotic open interacting systems. Besides free market economies, other examples of this path to cohesion are followed by Nature's systems and, perhaps unexpectedly, by some women's organizations. Women do not suffer from the excess of testosterone which pervades male organizations, requiring the male to assert his dominance. Instead, many women seem able to work co-operatively in groups without domination, some serving as co-ordinators to provide necessary and sufficient organization and infrastructure for stability.

We may use HUSH Principles to understand and discuss complex and sensitive situations, since the Principles and the Map offer a basis for objective analysis and synthesis. For example: -

  • If a (dominant) Premier were to eliminate from their Cabinet all members who did not share that Premier's views, then HUSH proposes that this suppression of variety will be followed by a degradation and break-up of the Cabinet
  • If a union of states were dominated by one country, with others held as vassals by terror, economic sanction, secret police, suppression of dissidents, etc., (stable by virtue of little variety but much connection) then any loosening of the grip by the dominant country would result in a domino break-up of the whole set, with the suppressed energy being released to generate new variety
  • After such a domino collapse, a plethora of political views and parties would generate, but would rapidly interact to produce many fewer, but larger and more powerful, sets as order emerged spontaneously from disorder
  • At the time of writing, the European Union is considering whether to open its doors to Eastern European Nations, formerly of the Soviet Union. From a HUSH perspective, the choice is straightforward. Stability of the European Union will be maintained and enhanced only of the variety represented by these new nations is well-connected to other members of the set, by virtue of cultural, social and economic interchange. If new members are not well-connected, then they are just as likely to act as dispersive influences as not
  • On the subject of the European Union again, HUSH suggests that attempts to unify the member states, to reduce their variety, to make them conform to some Euro-standard, will tend to reduce the stability of the whole. Connected variety is their potential strength
  • If a body is experiencing breakdown from dispersive influences, then the situation may be stabilized, not by control, but by the introduction of complementary systems which bind to the dispersive influence and to the remaining cohesive systems set. An ethnic minority which feels isolated and embattled may be brought into a more comfortable state by the introduction of a system for promoting the values of their culture to the other ethnic groups, by encouraging the adoption of a lower profile for those features of the culture causing most dispersion, and by cross-connecting the many varieties to their mutual benefit.
  • An information system comprising many distributed computers may be made more robust by employing an open systems interconnections and a variety of different processors and architectures. (While short-term economics might propose uniformity, HUSH promotes connected variety.) When change comes to the information system - new users, new needs, new applications, changing environment - the ability of the information system as a whole to adapt and evolve without interruption will be enhanced because some of the processing and software will already be suited to the change. A connected-variety system is not susceptible to unforeseen "systematic" weaknesses, as would be a system made from uniform processors and software. Hubble Space Telescope pointing hardware had to be replaced after a relatively short time in space owing to systematic errors in design which, as the components were replicated, affected all of them. Connected variety information systems are used for such essential applications as railway signalling, where positive and negative logic, together with alternate algorithms undertaking the same calculations, are used to ensure no single errors cause failures. Long-term economics favours connected variety, not uniformity

Notional Open, Interacting Systems Life-cycle

It would be useful to try some mathematical proof of HUSH, and that is both possible and underway. As a part of that "road to proof", models of system behaviour can be built. In the same way as Nature and Nurture were simulated previously, it is possible to build a simulation of Figure 71, and to run it to see what happens.

Figure 73. Expected results from simulating HUSH

Figure 73 shows what we might expect to happen, if the explanation of the HUSH model is anything to go by. In the figure, the Y-axis represents entropy, the degree of disorder. As varieties cohere, we might expect a drop in overall disorder. As dominance emerges in sets of systems, however, we would expect them to break up and eventually to reform into other variety groupings. Eventually, perhaps, we might expect a major decay as shown at the right of the figure, in the event of a reduction in energy to maintain the generation and mobility of variety. These predictions are all consistent with the Theory of Complexity. So what does the dynamic simulation really show?

HUSH Map Dynamic Simulation

In Figure 74, the y-axis is not entropy, but the number of cohesive systems within the overall number of systems at any time. Cohesive systems form a pattern in an otherwise random sea of varieties, so entropy goes down as we go up the Y-axis. The graph should therefore be upside down compared with Figure 73 - and that is broadly what happens. The general outline of Figure 74 is of the same shape as Figure 73 would suggest, but with the added feature that there is a rapid variation in the number of cohesive systems. It is as though stability is being achieved on two levels. At the higher level, there is a relatively slowly changing pattern, but when viewed more closely, there is a highly dynamic pattern. (The simulation assumes a constant energy inflow, so the tail at the right of Figure 73 is not expected to appear in Figure 74, and it does not.)

Such models are notoriously difficult to validate , and it would be quite wrong to interpret the simulation too freely, but the model does suggest chaotic behaviour in between the high spikes, which presage the break-up of cohesive groups as a result of dominance. Indeed, if the simulation is adjusted to avoid dominance, then a stable - but chaotic - state results. Stable, but chaotic? The number of cohesive systems still varies rapidly, but between limits which form smooth, horizontal contours. There are no paroxysms, suggesting - perhaps - that chaotic systems may be dynamically stable as expected.

Figure 74. Simulation of HUSH, as presented in Figure 71

Chapter 13


The Law Of Entropic Cycling

If you're anxious for to shine in the high aesthetic line as a man of culture rare

You must lie upon the daisies and discourse in novel phrases of your complicated state of mind,

The meaning doesn't matter if its complicated chatter of the transcendental kind

And everyone will say, as you walk your mystic way

"If this man expresses himself in terms too deep for me, Why, what a singularly deep young man this deep young man must be!"

W. S. Gilbert, 1836-1911

The Law of Entropic Cycling

HUSH from Chapter 12 provides the basis for conceiving and understanding the value and application of a new "Law of Complexity". What would such a new Law of Entropic Cycling be about? Like the Second Law of Thermodynamics, it will be about everything in general, but about nothing in particular. In fact, it should complement the Second Law; if that applies to closed systems, then the Law of Entropic Cycling should apply to open systems in a not-dissimilar way.

The following is proposed as a new Law of Entropic Cycling: -

Open, interacting systems' entropy cycles continually 
at rates and levels determined by available energy

The law applies to many, open, interacting systems, containing many systems in self-similar hierarchies, with the ordering mediated by connected variety. In this it is unusual, since it does not seek to operate within any boundary, unlike the Second Law which, by referring to isolated systems, implies a boundary across which energy does not pass in either direction. The Law of Entropic Cycling applies to a never-ending network of systems, providing a basis for understanding parts of this infinite network without bounds or preconceptions. Instead of isolation, the Law of Entropic Cycling embraces openness and interchange, with energy entering and leaving any part of the infinite network that may be of interest.

At first reading, the Law of Entropic Cycling may appear a disappointment. After all, how can it be used in connection with, say, a social system, an economic system, or an ecological system? First reading of the Second Law of Thermodynamics may engender similar reactions, but that Law is one of the cornerstones of physics and engineering. The Law of Entropic Cycling has to be seen in context. It applies to open, interacting systems which are in a state of continual, dynamic change. The Law is scale independent and type independent: it applies to large and small, systems of any kind. The Law proposes that systems, far from decaying with time, recycle themselves for ever, provided they receive energy continually.

How might we apply the Law? In essence, we already have done so, at least by example. The last chapter showed examples of HUSH analysis of complex situations; each is an example of the Law of Entropic Cycling in operation. For open, interacting systems, the Law of Entropic Cycling proposes that: -

  • Complexity can be measured as (configuration) entropy; the greater the variety, connection and tangling, the greater the disorder in pattern
  • Energy may take many forms including wealth, ideology, belief, and many more, according to the types of system of interest
  • A reduction in energy inflow to zero will result in zero entropy, eventually - consistent with the Second Law
  • An influx of energy will increase the general level of entropy, i.e. create more disorder - consistent with the Second Law
  • A steady flux of energy will result in continual entropy cycling, but not necessarily continuous entropy cycling, i.e. the rate of change of entropy need not be smooth
  • A rise in the energy flux to a new level will increase the rate of entropy cycling
  • There is a balance at all times between the energy influx and the total energy contained in, and utilized by, systems as they traverse the HUSH life cycle - conservation of energy applies for social and transcendental systems, too

Using these precepts, new viewpoints about social systems behaviour are possible concerning, for example, criminality and the law, education, committees, power structures, organizational behaviour, etc. Similarly, new viewpoints emerge about economic systems, engineering systems, transcendental systems, and even physical systems, particularly at quantum level for example.

Seeing Through the Dark Glass

We have come a long way since thinking about complexity as variety, connectedness and disorder. The thirteen chapters have painted on a wide canvas, looking at beliefs, psychology, ancient history, economics, politics, social science, genetics, heredity, Darwinian and social evolution, business, organization, behaviour and many, many more. The topics share in common that they represent complex ideas and situations which we may find interesting, but difficult to understand and interpret. While there are many theories to help us understand the particular in physics, astronomy, etc., there are few to help us understand the general, the whole. Biology and ecology, macro-economics and non-linear dynamics venture into these regions, and provide insights in so doing.

Fractal 13. Tunnel

At present, the West is still heavily caught up in Descartes reductionism. We see this philosophy in our schools, accounting, management, engineering and science, politics, social organization, etc. It gives us knowledge. But does it give us understanding? A visitor from another planet could gain knowledge about a clock, simply by taking it apart and seeing how it worked. That visitor might never understand the concept of time. Understanding comes from synthesis, from building up rather than breaking down, from looking outwards rather than inwards, from synergy rather than separation. A recent theory about the pyramids by the River Nile likened them to the Orion constellation and the Milky Way. That notion arose by looking outwards/upwards from the pyramids, not by looking inside them.

The Theory of Complexity, then considers the world without pulling it apart. Dismantling a clock is easier than, say, experimenting with a prison during a riot - in the real world, scientific experiment is often denied for very practical reasons. The Theory is one of hope for continuance, for improvement, rather than a theory of eventual decay, like the Second Law of Thermodynamics. The Theory does not predict the future, except in the senses of trends and continual recycling, but it might enable us to understand today and to influence our tomorrow. The search for a Theory of Everything is a noble venture and deserves to succeed. The Theory of Complexity might, just might, turn out to be more useful in everyday life.



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Fractal 14. Spirals



Last updated: Sep 2002