Exploring the science of complexitySystems & complexity

Exploring the science of complexity series (part 16): Concept 7 – Strange attractors and the ‘edge of chaos’

This article is part 16 of a series of articles featuring the ODI Working Paper Exploring the science of complexity: Ideas and implications for development and humanitarian efforts.

Complexity and systems – Concepts 4, 5, 6, and 7

The next four concepts relate to different aspects of how complex systems – those characterised by Concepts 1-3 – change over time. The causal relationships that play out within complex systems are explained using the concept of nonlinearity (Concept 4) and the sensitivity of complex systems to their starting conditions is highlighted (Concept 5). The overall shape of the system and its future possibilities are described using the idea of phase space (Concept 6). The patterns underlying seeming chaos within complex systems are explained (Concept 7).

Concept 7 – Strange attractors and the ‘edge of chaos’

Outline of the concept

The concept of phase space and attractors are central to understanding complexity, as complexity relates to specific kinds of system trajectories through phase space over time. The behaviour of complex systems can at first glance appear to be highly disordered or random. Moreover, these systems move through continually new states, with change as a constant in a kind of unending turbulence. However, there is an underlying pattern of order that is recognisable when the phase space of the system is mapped, known as a strange attractor.

Detailed explanation

In the 19th century, a mathematician named Henri Poincare was using Newton’s equations of planetary motion, which were – as has already been covered – based on a number of assumptions of linearity. Poincare proved that this approach worked for simple planetary systems of two bodies. In order to test the applicability with systems of three bodies – e.g. the sun and two planets – Poincare used tools that were based on the same principles as phase space to map the movement of such systems over time. He found the trajectory of the system to be one of ‘awesome complexity’ 1. The idea lay more or less dormant until the 1960s, when the Lorenz experiments showed that computer-aided modelling could be used to identify complexity.

Until the 1960s, there were only a few known attractors – including fixed points and periodic (as described previously in concept 6). All of these attractors related to systems that are predictable, in terms of understanding where they may end up. However, complex systems that are hard to predict also have an attractor, but they are much harder to map without the use of computers. The attractor for complex systems was discovered by Lorenz (shown in Figure 1). Most commonly known as strange attractors2, these are at the heart of the understanding of complexity.

Strange attractors show how complex systems move around in phase space, in shapes which resembles two butterfly wings3. A complex system – such as the three-body planetary system, or the weather – would move around one loop of the attractor, spiralling out from the centre. When it got close to the edge of the ‘wing’ it would move over to the other ‘wing’ and spiral around again4. Complex systems can have a chaotic dynamic, and develop through a series of sudden jumps5. Such a jump, usually referred to as a bifurcation, is an abrupt change in the longterm behaviour of a system, when the value of a particular dimension becomes higher or lower than some critical value. As one gets close to the bifurcation points – which may be seen as those points where the system moves from one wing of the attractor to the other, the values of fluctuations increase dramatically.

This strange attractor shows that complexity – although seemingly completely disordered, actually displays order at the level of its trajectory, and that although it may be unpredictable in its detail, it always moves around the same attractor shape. This ‘narrowness of repertoire’ is at the heart of the order hidden in complexity.

The strange attractor
Figure 1. The strange attractor (source: adapted from Mendenhall et al. 19986).
The lines in the attractor reflect the overall pattern of system behaviour, rather than the sequential movement of the system through time. Points on this attractor appear haphazardly at various locations on the lines, over time, eventually revealing the lines, but giving the observer no clue as to where the point will next appear within phase space. Eventually, the overall pattern of system behaviour is revealed. As Sanders puts it:

There are systems that never settle into a predictable or steady state … these are said to have strange attractors. A graphic representation of such a system will reveal a complicated pattern or shape, where the internal design never repeats itself 7 8

These observations offer an explanation of why elaborate computer programs cannot predict weather patterns with 100% accuracy. Yet, although the weather is unpredictable, it remains bounded within a certain ‘space of the possible’. A complex system is thus dynamic and nonlinear, and it is hard to predict the outcome of a given input and the feedback loops this causes. When the feedback is positive there is progression: the system moves forward. Feedback loops do not always produce the same effects and are not predictable. Paradoxically, complex feedback systems act to control the chaos in complex systems and keep them within certain boundaries9. A somewhat poetic view of a city from this perspective illustrates this:

Buyers, sellers, administrations, streets … are always changing, so that a city’s coherence is somehow imposed on a perpetual flux of people and structures. Like the standing wave in front of a rock in a fast-moving stream, a city is a pattern in time. No constituent remains in place but the city persists10

Such systems do display order, albeit not in the regular sense expected with linear systems. Instead, the order relates to the shape or pattern that the behaviour of a system displays in its phase space over time.

At a more general level, the notion of strange attractors and bifurcations implies that, despite chaotic or turbulent behaviour, the dynamics of complex systems can be investigated and understood. With the use of these tools, complexity scientists have been able to shed light on situations where there is no settling down to a stable equilibrium, no stable states and no repetition. Instead, there are systems undergoing continuous change, driven by the various factors and actors that shape and make them. This process of continuous change is often referred to as far from equilibrium, or ‘unending turbulence’.

This resonates with much thinking in political science, which suggests that ‘economic innovation [is] often driven by social conflicts within economic systems [and] seems to be a constant generator of fluctuations in capitalist social systems’ 11. In fact, it is possible that a very large number of phenomena in the physical and social worlds can be better understood as complex systems undergoing continuous change and operating far from equilibrium. To cite one thinker [Jake Chapman], ‘our society and all of its institutions are in continuing processes of transformation … we must learn to understand, guide, influence and manage these transformations’ 12. For some, this means operating at the ‘edge of chaos’. A previous ODI working paper looks at applying complexity theory to the process of strengthening capacity in community-based natural resource management organisations 13. Using examples of organisations from the Fiji Islands, Papua, West Bengal and Venezuela, Warner examines three approaches to the adaptation of CBNRM and investigates how organisations can be assisted to manage and adapt in the face of these increasing development pressures. Warner argues that, within certain limits,

… methods of interest-based negotiation can be applied to solicit organisation-specific rules that draw … organisations away from development-induced conflict and social exclusion towards an “edge of chaos” where creativity and adaptation flourish14

Box 1 provides a more detailed look at the concept of ‘edge of chaos’, with specific reference to urban planning and urban regeneration.

Box 1: Edge of chaos

Attractors suggest that systems are understood in terms of the two extremes of order and chaos. The metaphor of solids and gases can be used to clarify this. In solids, atoms are locked into place, whereas in gases they tumble over one another at random. However, right in between the two extremes, at a phase transition, a phenomenon called the ‘edge of chaos’ occurs. This phenomenon describes systems behaviours where the components of the system never quite lock into place and never quite dissolve into turbulence either. In human organisations, the simplest example is of a system that is neither too centrally controlled (order) nor too unorganised (chaos). The key question for many thinkers, who suggest that the edge of chaos is the place of maximum innovation within human systems, is how complex systems get to the edge of chaos. The illustration above on solids and liquids suggests, logically, that ordered systems can achieve this by loosening up a bit, and chaotic ones can do it by getting themselves a little more organised.

A study of urban planning has suggested that the edge of chaos principle relates to the evolving relationship between local authorities and local communities in Hulme, Manchester. This shaped decision-making processes, steering a path between the two extremes of centralised order (local authorities) and bottom-up chaos (community groups). Using social network analysis, the diminishing gap between authorities and communities was measured, drawing conclusions about the strength of the ties and frequency of interaction between the two groups over time. It was found that, from 1960–85, decision making was enshrined in the notion of local authorities making decisions for local communities without the latter being consulted. By the mid-1980s, consensus was beginning to loosen up under the exigencies of the (emergent) local community networks, moving from the highly centralised ‘we know best’ spirit of the 1960s to an acceptance of the opinions of community groups as useful and valuable in the decision-making process. This represented a massive change and paved the way for real progress in Hulme in the 1990s. Hulme as a system was searching for the edge of chaos, a special kind of balance (in decision making) between central control and the power of community networks. An important point to note here is that nobody designed the search process for the consensus that ensued – the system itself found the balance. A programme was then launched which was able to flourish on this fertile ground of strengthened community–authority interactions. Subsequent evaluations on the regeneration processes highlighted that the success of the initiative came about because it was at a particularly innovative point for community–authority relations. This highlights a potentially fundamental insight into the understanding of the urban system in general and urban regeneration processes in particular.

Source: Moobela 200515.

If social systems cannot best be described by reference to fixed point or periodic attractors, this means that social phenomena should not be viewed as tending towards equilibrium, as having defined endstates, or as being cyclical. A more apt metaphor, and one which may help to further understanding, may be to view them as open systems that exchange energy, matter or information with each other and their environment, and that continually create new structures and order 16. Nobel Prize winner Ilya Prigogine has referred to these as ‘dissipative structures’, in reference to his research on a wide range of systems that displayed such behaviours. Such systems can maintain themselves in stable states which are far from equilibrium, and can transform themselves into new structures of increased complexity. Prigogine’s analysis details how instabilities and bifurcations to new structures are in fact the result of fluctuations which are amplified by positive feedback processes.

Example: Organisational change as bifurcations in a complex system

Gareth Morgan17 suggests that the idea of strange attractors provides a powerful perspective for the management of stability and change in organisations. Specifically, he suggests that transformational change ultimately involves the creation of new contexts that can break the hold of dominant attractor trajectories in favour of new ones. He uses the idea of a strange attractor as a creative metaphor (as shown in Figure 2) to generate thinking about organisational change, and in doing so raises an important challenge for managers of change processes. If organisations can be described using the attractor metaphor, then it is implied that managers cannot be in control of the change. The new pattern of the attractor cannot be precisely defined – it is only possible to nurture elements of the new context, and create conditions under which the new context can arise. When the old pattern – the old context – is particularly powerful, no significant change is possible, because the organisation ends up trying to do new things in old ways. Morgan sees that the power of this approach lies in its potential both to open up new understandings and possibilities for action but also, importantly, to outline the limitations in terms of individual actors’ control and power over organisational change processes.

A strange attractor as a creative metaphor
Figure 2. A strange attractor as a creative metaphor (source: Morgan 198618).

Implication: Manage contexts and ensure decision-making approaches are appropriate to the system

Certain social, economic and political domains seem – at least metaphorically – to fit the image of the strange attractor metaphor, with discontinuities, perpetual novelty and ever-changing elements but recurring patterns and discernible structures. This is true of international aid – as Porter et al.19 argue, the whole international development system is ‘a moving, evolving multi-faceted thing, and if it was possible to offer an answer today, it would be inappropriate by tomorrow.’

This equally applies to the process of development in a country, or the adequate means of responding to crises. A particular issue in international aid may not be most usefully solved through the provision of a particular ‘output’. Rather, it may be more productive to see development as an open-ended, ongoing, unpredictable and continually changing process. Similarly, crises can be seen as bifurcation points in which human social systems are exposed to high constraints and stress ‘that upset the balance between the internal forces structuring the systems and the external forces that make up the environment’ 20. A crisis could then be defined as a condition in which there is a change in an environmental or human stress that is destabilising enough so that the original set of attractors is supplanted by a new set of attractors.

However, this perspective contrasts with attitudes towards chance and risk prevalent within aid organisations:

Venturing into the unknown normally means that the organisation’s standard operating procedures can no longer deal with the types of information it is receiving, and are no longer suitable. Such departures occur when the organisation is on the brink of collapse or is being forced – by means no longer in its control – to change its procedures fundamentally. It often takes a long time for an [aid] organisation to realise that it has hit the point where there is no alternative to change; often, that point comes too late21

Accepting the notion of chaos and strange attractors encourages an acknowledgement of the continual change in social systems, which by extension requires acceptance of ‘the inevitability of change’ in the many systems that aid agencies operate within and around. Such change should not be viewed as worrying or necessarily negative22. Equally, equilibrium and stability should not be viewed as default and ideal states for a particular system, but as situations of stasis and ossification. Incorporating this insight into the way problems are approached in the development sector points towards an important shift in thinking. As Peter Senge suggests:

… most of us have been conditioned throughout our lives to focus on things and to see the world in static images. This [in turn] leads to linear explanations of “systemic phenomenon”. Understanding the perpetual flux in systems should lead us to see “interrelationships, not things, and processes, not snapshots”…23

Whether looking within aid organisations or outside them, this calls for better management of context, and to give up the idea of precise control in favour of the idea of the emergent nature of change. New contexts can be generated through new understandings, such that those operating in the system can be encouraged to challenge and change existing paradigms, norms and assumptions. For example, thinking of an organisation as discrete may lead those within it to try and help it survive as a discrete entity, instead of allowing it to evolve to a new form. New contexts can also be encouraged by identifying and changing the ‘basic rules’ which reinforce the existing attractor patterns, allowing new actions to emerge and become powerful messages for the kind of change that is being sought. These changes can help to catalyse other changes that are in line with the hoped-for new context24.

The potential for small changes to lead to large, directed changes is explored by Holland in his work on ‘lever points’ of systems. Holland25 argues that such lever points could be the key to solving problems such as ‘immune diseases, inner city decay, industrial innovation, and the like’. The chaos metaphor suggests that there are bifurcation points that tip systems from one state to another – and, if these can be understood, then it may be possible to better identify such leverage points. This has been explored in the context of aid effectiveness26 by investigation of the premise that a relatively small intervention through small grants and technical cooperation assistance might cause a disproportionably significant impact. The study suggests that donors may already be funding these kinds of ‘high leverage’ initiatives, but that current reporting procedures and increasing interest in large-scale budget support may mean that these activities and the factors that might contribute to their success are not well documented. These factors are likely to include: the country-specific context; the approach of the donor agency to aid effectiveness; how it understands change; how it invests in relationships; its openness to a diversity of views; and its preparedness to experiment, take risks and learn to alter its views. All of these can be seen as ways to manage the context within which change happens.

The inherent unpredictability of change in such systems also means that there may be significant value in an organisation increasing its ‘agility’. In systems characterised by ‘surprise and discontinuity … organisations need to rapidly adapt to unexpected conditions … they have to improvise27. Thinking back to concept 5 and the implications for planning, this does not – as some claim – imply that strategy becomes irrelevant: ‘the idea of strategising for the future is fundamentally based on the unpredictability of the future, of which some aspects … can be foreseen’. To put it another way, working with chaos means ‘it is not about being strategic or opportunistic; it is about being strategically opportunistic’ (John Young, personal communication). It has been suggested that ‘the adoption of minimally structured organisational forms are a necessary condition for strategic improvisation’ 28.

These factors also resonate with suggestions made for dealing with continuous transformation and change. Specifically, some see the potential turbulence of chaotic systems as emphasising the high importance of making continuous learning an inherent part of organisations and policy. To continue the quote from Jake Chapman cited … [above]:

Our society and all of its institutions are in continuing processes of transformation … we must learn to understand, guide, influence and manage these transformations. We must make the capacity for undertaking them integral to ourselves and our institutions. We must, in other words, become adept at “learning”. This “learning” should not be seen as a one-off event, or a case of acquiring new knowledge or skills, rather it involves ongoing practice and reflection on one’s own experience. Since knowledge of “best practice” cannot be easily imported from elsewhere, all organisations must involve themselves in learning as a “continuous, on-the-job process” 29

This should be done through a commitment to ongoing reflection and adaptation of aid programmes. Since the context in which a programme is operating is continuously changing, and it is not possible to plan for all eventualities, a successful programme is one that assesses and adapts to changing situations in an intelligent way based on thoughtful reflection. This means that the programme needs to be engaged in ongoing reflection and learning so as to remain relevant and appropriate. This shift towards ongoing processes of learning has some knock-on implications. It has been suggested that this entails a move in attitude away from ‘knowing best’ (‘if one already knows the answer or knows best then there is no need to learn anything’)30, realising that there are ‘no final answers’ and we must approach problems with the mindset of ‘enquiry and not certitude’ 31. This could shift focus of policy from ‘specifying targets to be met’ towards ongoing work ‘based on learning what works, and towards improving overall system performance, as judged by the end-users of the system’ 32. But a chaotic system not only suggests that lessons themselves are permanently provisional33, but also calls for an approach to learning and decision making that is tailored to the specific situation.

Although it is tempting to suggest methodologies that enable this kind of thinking to be implemented, such as soft systems methodology, or outcome mapping, or most significant change, in reality this reinforces the notion that tools are useful but no single tool should be expected to provide all of the guidance needed for decision making. Similarly, no single tool should be expected to provide the most appropriate means by which to arrive at guidance. The notion of strange attractors and chaos goes further and suggests that no single mindset should be seen as the appropriate to all settings.

Researchers from IBM [Kurtz and Snowden]34 have done interesting empirical studies which relate to different kinds of organisational systems, with careful attention paid to those which feature chaotic dynamics. Kurtz and Snowden characterise certain decision-making and learning approaches as most appropriate to different kinds of systems. For example, approaches that focus on sensing incoming data, categorising it and responding in accordance with established practice are most appropriate in systems that are ordered and known, for example, when undertaking business process re-engineering, in which cause and effect relationships are seen as linear and understood. Examples of such approaches are single-point forecasting, field manuals and operational procedures.

By contrast, approaches that focus on sensing data, analysing it and then responding in accordance with expert interpretation and advice are most useful in complicated systems, for example, organisational learning initiatives, or strategic futures planning efforts, where there may be stable cause-and-effect relationships, and in which everything can be understood, given sufficient resources and time. Examples are experimentation, expert opinion, fact finding and scenario planning. While structured techniques are desirable, underlying assumptions must also be open to examination and challenge.

Finally, approaches that focus on sensing patterns of change and understanding multiple perspectives, and working to strengthen wanted patterns and weakening the unwanted are most appropriate in complex systems characterised by multiple feedback processes and interaction among many agents, emergent properties, nonlinear relationships and limited predictability. In such systems, many examples of which have been covered already, the application of structured methods will frequently confront new and different patterns for which they are not prepared, and approaches need to be tailored to the nature of the problem.

Next part (part 17): Concept 8 – Adaptive agents.

Article source: Ramalingam, B., Jones, H., Reba, T., & Young, J. (2008). Exploring the science of complexity: Ideas and implications for development and humanitarian efforts (Vol. 285). London: ODI. (https://www.odi.org/publications/583-exploring-science-complexity-ideas-and-implications-development-and-humanitarian-efforts). Republished under CC BY-NC-ND 4.0 in accordance with the Terms and conditions of the ODI website.

Header image source: qimono on PixabayPublic Domain.

References and notes:

  1. Capra, F. (1996). The Web of Life, London: Flamingo/Harper Collins.
  2. But also referred to as chaotic, Lorenz and butterfly attractors
  3. Given the use of the butterfly metaphor by Lorenz, some have assumed that there is some connection between the two. However, it seems as though this is just a rather happy coincidence.
  4. Strogatz, S. (2003). Sync: The Emerging Science of Spontaneous Order, London: Penguin Books.
  5. Feigenbaum, M. (1978). ‘Quantitative Universality for a Class of Nonlinear Transformations’, Journal of Statistical Physics 19(1), 25–52.
  6. Mendenhall, M., Macomber, J., Gregersen, H. and Cutright, M. (1998). ‘Nonlinear Dynamics: A New Perspective on IHRM Research and Practice in the 21st Century’, Human Resource Management Review 8(1): 5–22.
  7. Sanders, I. (1998). Strategic Thinking and the New Science: Planning in the Midst of Chaos, Complexity, and Change, Columbus OH: The Free Press.
  8. If any part of the strange attractor were magnified, it would reveal a multi-layered sub-structure in which the same patterns are repeated. Complexity plays out in identical ways at different levels of a system. The development of fractal geometry by the IBM researcher Benoit Mandelbrot has helped to further understanding of chaos, to the extent that the term ‘fractal’ is now widely used to describe the computer-generated images created when mapping strange attractors (Gleick, J. (1987). Chaos: Making a New Science, New York: Viking.)
  9. Nilson, T. H. (1995). Chaos marketing: How to win in a turbulent world, London: McGraw-Hill.
  10. Holland, J. (1995). Hidden Order: How Adaptation Builds Complexity, New York: Helix Books.
  11. Byrne, D. (1998). Complexity Theory and the Social Sciences: An Introduction, London: Routledge.
  12. Chapman, J. (2004). System Failure: Why Governments Must Learn to Think Differently, London: Demos.
  13. Warner, M. (2001). Complex Problems … Negotiated Solutions: The Practical Applications of Chaos and Complexity Theory to Community-based Natural Resource Management, London: ODI.
  14. Warner, M. (2001). Complex Problems … Negotiated Solutions: The Practical Applications of Chaos and Complexity Theory to Community-based Natural Resource Management, London: ODI.
  15. Moobela, C. (2005). ‘From Worst Slum to Best Example of Regeneration: Complexity in the Regeneration of Hulme, Manchester’, E:CO 7(1): 29–42.
  16. Mittleton-Kelly, E. (2003). ‘Ten Principles of Complexity and Enabling Infrastructures’ in Complex Systems and Evolutionary Perspectives of Organisations: The Application of Complexity Theory to organizations, London: Elsevier Press.
  17. Morgan, G. (1986). Images of Organization, 3rd edition updated in 2006 London: Sage.
  18. Morgan, G. (1986). Images of Organization, 3rd edition updated in 2006 London: Sage.
  19. Porter, D., Allen, B. and Thompson, G. (1991). Development in Practice: Paved with Good Intentions, London: Routledge.
  20. Laszlo, E. (1991). The age of bifurcation: Understanding our changing world, New York: Gordon & Breach.
  21. Kent, R. (2004). ‘Humanitarian Futures: Practical Policy Perspectives’, Network Paper 46 (April). London: ODI/Humanitarian Practice Network.
  22. Haynes, P. (2003). Managing Complexity in the Public Services, Berkshire: Open University Press.
  23. Senge, P. (1990). ‘The Leader’s New Work: Building Learning Organizations’, Sloan Management Review 32(1): 7–23.
  24. Morgan, G. (1986). Images of Organization, 3rd edition updated in 2006 London: Sage.
  25. Holland, J. (1995). Hidden Order: How Adaptation Builds Complexity, New York: Helix Books.
  26. Eyben, R. (ed.) (2006). Relationships for Aid, London: Earthscan.
  27. Pina e Cunha, M. and Vieira da Cunha, J. (2006). ‘Towards a Complexity Theory of Strategy’, Management Decision 44(7).
  28. Pina e Cunha, M. and Vieira da Cunha, J. (2006). ‘Towards a Complexity Theory of Strategy’, Management Decision 44(7).
  29. Chapman, J. (2004). System Failure: Why Governments Must Learn to Think Differently, London: Demos.
  30. Chapman, J. (2004). System Failure: Why Governments Must Learn to Think Differently, London: Demos.
  31. Westley, F., Zimmerman, B. and Quinn Patton, M. (2006). Getting to Maybe: How the World is Changed, Toronto: Random House.
  32. Chapman, J. (2004). System Failure: Why Governments Must Learn to Think Differently, London: Demos.
  33. Chambers, R. (1997). Whose Reality Counts: Putting the First Last, London: Intermediate Technology Publications.
  34. Kurtz, C.F. and Snowden, D. (2003). ‘The New Dynamics of Strategy: Sense-making in a Complex and Complicated World’, IBM Systems Journal 42(3).
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Ben Ramalingam and Harry Jones with Toussaint Reba and John Young

Authors of the Overseas Development Institute (ODI) Working Paper "Exploring the science of complexity: Ideas and implications for development and humanitarian efforts".

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One Comment

  1. Is there any software that enables us to show, better manipuate, strange attractors as for example how one engulfs another?

    Best, Chris

    Chris Warren-Adamson PhD
    University of Southampton retired.

    Subject: how complexity theory helps us to understand interventions and processes in child welfare

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