8 Mar 2010

Forks & Dice: Bifurcation in Prigogine & Stengers, Order out of Chaos: Man's New Dialogue with Nature

by Corry Shores
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Forks & Dice:
Bifurcation in Prigogine & Stengers,
Order out of Chaos: Man's New Dialogue with Nature
La Nouvelle Alliance: Métamorphose de la science

In the second section of the third chapter in Cinema 2, Deleuze describes a kind of forking where a path of development skews-off wildly at unstable points:

And it is not just the circuits forking between themselves, it is each circuit forking within itself, like a split hair. [...] its repetitions are not accumulations, its manifestation refuse to be aligned, or to reconstitute a destiny, but constantly split up any state of equilibrium and each time impose anew 'meander', a new break in causality, which itself forks from the previous one, in a collection of non-linear relations. [footnote 8: On this notion of forking, cf. Prigogine and Stengers, Order out of Chaos: man's new dialogue with nature, London: Heinemann, 1984, pp. 189-90.] [Deleuze 47bc; 47c; 280a]

Et ce ne sont pas seulement les circuits qui bifurquent entre eux, c' est chaque circuit qui bifurque avec soi-même, comme un cheveu four-chu. [...] ses répétitions ne sont pas des accumulations, ses manifestations ne se laissent pas aligner, ni reconstituer un destin, mais ne cessent de morceler tout état d'équilibre, et d'imposer chaque fois un nouveau « coude », une nouvelle rupture de causalité, qui bifurque elle-même avec la précédente, dans un ensemble de relations non-linéaires [note 7: Sur cette notion de bifurcation, cf. Prigogine et Stengers, La nouvelle alliance, Gallimard, p. 190]. [Deleuze 68d; 69a; 69d]

We will avoid most of what is difficult about bifurcation, even though it is essential for fully grasping the concept. We do so on account of your author's limitations. I probably know less than most readers, who are invited to make corrections and provide better explanations. But also it seems the basic ideas that we need to apply in the context of cinema can be presented in a highly simplified form. So that is what we hope to accomplish in the following.

We will first discuss the notions of linearity and non-linearity, drawing from this page at Mathematics Illuminated.

Consider first x = 2.

And x = 2y

We notice that there is one line that tends the same direction throughout.

The equation is "linear" because its graph (all the "x,y" points on the coordinate plane that satisfy the equation) is a straight line, and also because a small change in the value of x effects a proportional, constant change in y. ("Linear vs. Nonlinear Systems")

But now consider x-squared = 4

And now also x-squared = y

We notice two things about the diagram immediately above. The higher power caused there to be a bifurcation of values. And the lines they graph are not straight or 'linear'.

A nonlinear equation is something that doesn't have just a first power of the independent variable and consequently can't be graphed as a simple straight line. ("Linear vs. Nonlinear Systems")

Mathematics Illuminated then gives the example of pendulum motion to illustrate the difference between linear and non-linear systems.

These so-called nonlinear systems can exhibit some wild behaviors, behaviors that might be considered surprising, behaviors that don't fit so nicely into equations. For example, our simple pendulum behaves very smoothly and predictably as long as it doesn't swing too high.
For larger and larger angles, the range of possible behaviors is more varied than the simple cycling back and forth. For example, if the pendulum has sufficient momentum, it will swing past the horizontal line of the pivot and go all the way around, over the top. If it has a little less momentum than this, it might stall near the vertical position above the pivot, lose the tension of the string, and drop almost straight down under the influence of gravity. Both of these behaviors are examples of nonlinearities. ("Linear vs. Nonlinear Systems," emphasis mine)

(Image obtained gratefully from Mathematics Illuminated)

For the sake of illustration, we might imagine that the pendulum can reach a critical point where there are a number of outcomes, none of which can be predicted. We might then again for the sake of illustration regard there to be a bifurcation point where the development of the system can fork-off into very different directions. Perhaps the slightest infinitesimal fluctuation can cause profoundly different outcomes.

In Order out of Chaos: Man's New Dialogue with Nature (La Nouvelle Alliance: Métamorphose de la science), Prigogine & Stengers describe bifurcation in chemical reactions. Hopefully we can profoundly simplify their explanation without falsifying it too much. The system will be stable if the variables are kept within certain bounds. But when an independent variable is pushed to a critical chaotic point, the dependent variable can veer-off or fork-away into two possible directions of development. They write:

Consider the bifurcation diagram represented in Figure 11.

This differs from the previous diagram in that at the bifurcation point two new stable solutions emerge. Thus a new question: Where will the system go when we reach the bifurcation point? We have here a "choice" between two possibilities; they may represent either of the two nonuniform distributions of chemical X in space, as represented in Figures 12 and 13.

The two structures are mirror images of one another. In Figure 12 the concentration of X is larger at the left; in Figure 13 it is larger at the right. How will the system choose between left and right? There is an irreducible random element; the macroscopic equation cannot predict the path the system will take. Turning to a microscopic description will not help. There is also no distinction between left and right. We are faced with chance events very similar to the fall of dice. (162-163, emphasis mine)

They write a bit later:

If we consider Figure 17 [...] we see that the system already has a wealth of possible stable and unstable behaviors.

The "historical" path along which the system evolves as the control parameter grows is characterized by a succession of stable regions, where deterministic laws dominate, and of instable ones, near the bifurcation points, where the system can "choose" between or among more than one possible future. Both the deterministic character of the kinetic equations whereby the set of possible states and their respective stability can be calculated, and the random fluctuations "choosing" between or among the states around bifurcation points are inextricably connected. This mixture of necessity and chance constitutes the history of the system. (169-170, emphasis mine)

Soon we will discuss the bifurcations of Mankiewicz' movies. When we do so, we will see that we arrive at critical and unstable points in the narrative where a character forks or bifurcates unpredictably. Prigogine & Stengers' bifurcation diagrams will illuminate this concept.

Pendulum image and direct quotations regarding linear and non-linear systems obtained gratefully from:

Linear and non-linear graphs made using the following freeware:
GIMP, and

The Prigogine & Stengers text citations and images from:

Prigogine, Ilya, and Isabelle Stengers. Order out of Chaos: Man's New Dialogue with Nature. London: Heinemann, 1984.

Prigogine, Ilya, and Isabelle Stengers. La Nouvelle Alliance: Métamorphose de la science. Paris: Éditions Gallimard, 1979.

Deleuze citations from:

Deleuze, Gilles. Cinema 2: The Time Image. Transl. Hugh Tomlinson and Robert Galeta. London & New York: 1989.

Deleuze, Gilles. Cinéma 2: L'image-temps. Paris: Les éditions de minuit, 1985.

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