introduction to nonlinear systems : : ian clothier

The development of modern nonlinear dynamics is generally acknowledged to have commenced with Poincare, in particular work completed between 1880 and 1910. Unfortunately, scientists largely ignored this work for half a century, although during this time mathematicians extended Poincare’s work, proving a range of his conjectures (Jackson, 1991).

Perhaps Poincare’s most significant work related to the study of the ‘three body problem.’ Isaac Newton had worked out the equations for the two-body (planets) problem – say, the movement of the earth and moon. It was considered that the universe was a clockwork mechanism; all that was required was sufficient knowledge of the workings of the mechanism. Poincare showed that the three-body problem is fundamentally unsolvable: that when three bodies or planets are considered, the instabilities are such that accurate prediction is no longer possible.

In describing the character of nonlinear equations, Jackson (1991, p.6) wrote that “the ratio (action/reaction) is not constant.” This is the same as saying cause and effect has broken down. One cause could at another time produce a different effect. In nonlinear systems, the effect can feedback into the cause, generating not just a fresh result, but also a new context for interaction. Jackson wnet to write (ibid.):

Nonlinear phenomena concern processes involving ‘physical’ variables, which are governed by nonlinear equations. These models have been obtained by some approximate ‘projection’ rationale from presumably more fundamental microscopic dynamics of the system.

The presence of nested (i.e. smaller scale) variables is explicit. Not only is the ratio of action to reaction not constant, forces within nonlinear systems cross energy/matter/natural/human boundaries. Thus De Landa (1997, p. 26-27) writes of the mineralization of elements in the sea forming basic skeletal structures, eventually leading to the development of the human endoskeleton, and later the mineralization of the human exoskeleton in the form of modern dwellings, enabled by clay and bricks.

Viewing all aspects of all systems – from cosmology to objects, acts and personality – as energies that propagate actualities, greatly assists understanding nonlinear processes. De Landa, (1997 p. 64, citing Deleuze and Guattari) writes of “an articulation of superpositions… an interconnection of diverse but overlapping elements.” Chemical elements and processes migrate into the human biological landscape; the same energy migrates onto the human cultural plateau.

At the boundary of overlapping parts of such “meshworks,” are “intercalary elements” which effect interconnections. The intercalary elements involve “densifications, intensifications, reinforcements, injections, showerings,” enabling congealment, expansion, diversification and compression. (De Landa 1997 p. 64, citing Deleuze and Guattari). It is notable that nonlinearity appears in both structural and post-structural theories.

It is perhaps easy to see why the ratio between action and reaction might not be constant in complex, self-organizing, adaptive systems[^1]. The articulation of superpositions across scales from the microcosmic to the macrocosmic – from chemical elements all the way to large-scale cultural systems, and the dynamics of interaction between these forces, creates both recurring and novel system states.

An aspect of nonlinear systems perhaps less easy to grasp is that nonlinearity is not mutually exclusive to linearity. Complex systems such as the weather are composed of linear steps. At many points in a storm, there is an increase in moisture content in the air. But to discuss a storm in terms of linear steps misses grasping the overall sense of the storm. The coherence enabled by a nonlinear view of processes captures the overall view. The process of collapsing large quantities of detail (such as the linear components of a storm) into a comprehensible view, is described by Douglas Hofstadter as ‘chunking.’

Chunking, determinism and unpredictability

Consider for a moment, the hierarchical levels of science occurring within the human body. To understand a human being, as Hofstadter (1979 p. 305) writes, we do not need to understand:

… the quark model, the structure of nuclei, the nature of electron orbits, the chemical bond, the structure of proteins, the organelles in a cell, the methods of intercellular communication, the physiology of the various organs of the human body, or the complex interactions among organs… Although there is some ‘leakage’ between… levels of science… there is almost no leakage from one level to a distant level. All that a person needs is a chunked model of how the highest level acts; and as we all know such models are realistic and successful.

Each level described contains a huge amount of information, and it is worth noting that these levels are active all the time, in every one of us. While Hofstadter notes that there is not much leakage between distant hierarchical levels, it could be inferred that what does leak to distant levels is important information.

In a chunked view, a sacrifice is made:

There is however one significant negative feature of a chunked model: it usually does not have exact predictive power… Despite not being sure how people will react to a joke, we tell it with the expectation that they will do something such as laugh, or not laugh – rather than, say, climb the nearest flagpole… A chunked model defines a ‘space’ within which behavior is expected to fall, and specifies probabilities of it’s falling in different parts of that space (Hofstadter 1979, p. 306).

The space Hofstadter implies would be equivalent to the attractor of the system.

Summary

An understanding of nonlinear systems has greatly altered the world-view of many people, in particular scientists, academics and thinkers, away from the linear and into a much broader awareness of relevant factors in describing the world. The metaphor of energies articulated across scales and over previously exclusive boundaries is opening up Western understanding.

Notes

[1]. Nonlinear systems such as the weather are governed by attractors, which generate fixed, cyclic or strange attractor states. When a state in a strange attractor is perturbed, the system can change dramatically – and the force resulting in the change can be either small (sensitive dependence on initial conditions) or large. The systems alters state. This reorganization is known as self-organization, the term used in caber theory and business studies (along with ‘complex adaptive systems’).

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References

Butz, M.R. (1997). Chaos and complexity: implications for psychological theory and practice. Washington: Taylor and Francis.

De Landa, M. (1997). A thousand years of nonlinear history. New York: Zone Books.

Hofstadter, D. (1979) Godel, Escher, Bach: an eternal golden braid. London: Penguin Books.

Jackson, E. Atlee. (1991) Perspectives of nonlinear dynamics (volume 1). Cambridge: Cambridge University Press, p6.

A good introduction to these issues is James Gleick’s Chaos (1988, London: Heinemann).