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.

Credits:

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

http://www.learner.org/courses/mathilluminated/units/13/textbook/02.php

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

Geogebra

GIMP, and

OpenOffice

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.

Also,

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.

Stalh's Vitalism. Prigogine & Stengers. Order out of Chaos / La Nouvelle Alliance

by Corry Shores
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[The following is quotation.]

Stalh's Vitalism

in

Prigogine & Stengers

Order out of Chaos: Man's New Dialogue with Nature

La Nouvelle Alliance: Métamorphose de la science

According to Stahl, universal laws apply to the living only in the sense that these laws condemn them to death and corruption; the matter of which living beings are composed is so frail, so easily decomposed, that if it were governed solely by the common laws of matter, it would not withstand decay or dissolution for a moment. If a living creature is to survive in spite of the general laws of physics, however short its life when it is compared to that of a stone or another inanimate object, it has to possess in itself a "principle of conservation" that maintains the harmonious equilibrium of the texture and structure of its body. The astonishing longevity of a living body in view of the extreme corruptibility of its constitutive matter is thus indicative of the action of a "natural, permanent, immanent principle," of a particular cause that is alien to the laws of inanimate matter and that constantly struggles against the constantly active corruption whose inevitability these laws imply [83-84]

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.

Entry Directory: Order out of Chaos / La Nouvelle Alliance by Prigogine & Stengers

by Corry Shores
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Entry Directory:

Prigogine & Stengers

Order out of Chaos: Man's New Dialogue with Nature

La Nouvelle Alliance: Métamorphose de la science

Instantaneous Tendencies

The Infinitesimal Straw that Broke the Camel's Back

Stalh's Vitalism

Forks & Dice: Bifurcation in Prigogine & Stengers

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.

The Infinitesimal Straw that Broke the Camel's Back. Prigogine & Stengers. Order out of Chaos / La Nouvelle Alliance

by Corry Shores
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[The following is quotation.]

The Infinitesimal Straw that Broke the Camel's Back

in

Prigogine & Stengers

Order out of Chaos: Man's New Dialogue with Nature

La Nouvelle Alliance: Métamorphose de la science

We have mentioned that trajectories correspond to deterministic laws; once an initial state is given, the dynamic laws of motion permit the calculation of trajectories at each point in the future or the past. However, a trajectory may become intrinsically indeterminate at certain singular points. For instance, a rigid pendulum may display two qualitatively different types of behavior - it may either oscillate or swing around its points of suspension. If the initial push is just enough to bring it into a vertical position with zero velocity, the direction in which it will fall, and therefore the nature of its motion, are indeterminate. An infinitesimal perturbation would be enough to set it rotating or oscillating. [...]

It is significant that Maxwell had already stressed the importance of these singular points. After describing the explosion of gun cotton, he goes to say:

In all such cases there is one common circumstance - the system has the quantity of potential energy, which is capable of being transformed into motion, but which cannot begin to be so transformed till the system has reached a certain configuration, to attain which requires an expenditure of work, which in certain cases may be infinitesimally small, and in general bears no definite proportion to the energy developed in consequence thereof. For example, the rock loosed by frost and balanced on a singular point of the mountain-side, the little spark which kindles the great forest, the little word which sets the world a fighting, the little scruple which prevents a man from doing his will, the little spore which blights all the potatoes, the little gemmule which makes us philosophers or idiots. Every existence above a certain rank has its singular points: the higher the rank, the more of them. At these points, influences whose physical magnitude is too small to be taken account of by a finite being, may produce results of the greatest importance. All great results produced by human endeavor depend on taking advantage of these singular states when they occur." [footnote 14. MAXWELL J.C., Science and Free Will, in CAMPBELL L. and GARNETT W., op. cit., p. 443. L. CAMPBELL & W. GARNETT , The Life of James Clerk Maxwell (London, Macmillan, 1882).] [73b.d/320c, emphasis mine]

Ainsi, des chercheurs notèrent qu'une trajectoire peut devenir intrinsèquement indéterminée en certains points singuliers. Un pendule rigide peut avoir deux types de comportements qualitativement différents : il peut soit osciller soit tourner autour de son point de suspension. Si son impulsion initiale est exactement suffisante pour le faire aboutir avec une vitesse nulle en position verticale, la direction vers laquelle il retombera, et donc la nature de son mouvement, est indéterminée : une perturbation infinitésimale suffit à entraîner soit une rotation soit une oscillation. [...]

Il est remarquable de constater que Maxwell avait déjà souligné l'importance de tels points singuliers : « Dans tous les cas de ce genre (Maxwell vient de décrire l'explosion du coton fulminant), il y a une circonstance commune : le système possède une quantité d'énergie potentielle qui peut être transformée en mouvement mais ne peut commencer à l'être que lorsque le système a atteint une certaine configuration, ce qui nécessite une dépense de travail, qui peut être infinitésimale et est en général sans commune mesure avec l'énergie qu'elle permet de libérer. Ainsi, le rocher détaché par le gel et en équilibre sur un point singulier du flanc de la montagne, la petite étincelle qui embrase l'immense forêt, le petit mot qui met le monde en guerre, le petit scrupule qui empêche l'homme de faire ce qu'il veut, le petit spore qui gâte toutes les pommes de terre, la petite gemmule qui fait de nous des philosophes ou des idiots. Chaque existence à partir d'un certain niveau a ses points singuliers : plus élevé le niveau, plus nombreux les points. En ces points, des influences, dont la taille physique est trop petite pour être prise en compte par un être fini, peuvent produire des résultats de la plus grande importance. Tous les grands résultats produits par les entreprises humaines dépendent de la manière dont on prend avantage de ces états singuliers, lorsqu'ils se présentent.» [note I. MAXWELL J.C., Science and Free Will, in CAMPBELL L. et GARNETT W., op. cit., p. 443] [85b-86]

[Also see the section on catastrophe theory towards the end of this entry.]

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.

Instantaneous Tendencies in Prigogine & Stengers Order out of Chaos / La Nouvelle Alliance

by Corry Shores
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[Central Entry Directory][Other entries in the Order out of Chaos / La Nouvelle Alliance series.]

[The following is quotation.]

Instantaneous Tendencies

in

Prigogine & Stengers

Order out of Chaos: Man's New Dialogue with Nature

La Nouvelle Alliance: Métamorphose de la science

How can a continuously varying speed be defined? How can we describe the instantaneous changes in the various quantities, such as position, velocity, and acceleration? How can we describe the state of a body at any given instant? To answer these questions, mathematicians have introduced the concept of infinitesimal quantities. An infinitesimal quantity is the result of a limiting process; it is typically the variation in a quantity occurring between two successive instants when the time elapsing between these instants tends toward zero. In this way the change is broken up into an infinite series of infinitely small changes.

At each instant the state of a moving body can be defined by its position r, by its velocity v, which expresses its "instantaneous tendency" to modify this position, and by its acceleration a, again its "instantaneous tendency," but now to modify its velocity. Instantaneous velocities and accelerations are limiting quantities that measure the ratio between two infinitesimal quantities: the variation of r (or v) during a temporal interval Δt, and this interval Δt when Δt tends to zero. Such quantities are "derivatives with respect to time," and since Leibniz they have been written as v = dr/dt and a = dv/dt. Therefore, acceleration, the derivative of a derivative,

,

becomes a "second derivative." (57-58b)

Comment décrire une vitesse qui varie de manière continue? Comment décrire l'évolution, d'instant en instant, des diverses grandeurs, positions, vitesse, accélération, qui caractérisent l'état instantané d'un mobile? Les mathématiciens ont introduit le concept de quantité infinitésimale pour répondre à de telles questions. Une quantité infinitésimale résulte d'un passage à la limite, c'est la variation d'une grandeur entre deux instants successifs lorsque l'intervalle entre ces deux instants tend vers zéro. La description infinitésimale peut ainsi décomposer le changement en une série infinie de changements infiniment petits, alors que, précédemment, on ne pouvait le décrire que comme le résultat d'un nombre fini de transitions de grandeur finie juxtaposées comme les perles d'un collier.

En chaque instant la description de l'état d'un mobile comprend non seulement sa position, que nous noterons r, mais encore sa « tendance instantanée » à changer de position, c'est-à-dire sa vitesse v en et instant, et sa tendance à modifier cette vitesse, c'est-à-dire son accélération a. Vitesse et accélération instantanées sont des « concepts limites » mesurant une variation instantanée comme le rapport entre deux quantités infinitésimales : la variation de la grandeur position ou vitesse, pendant un intervalle de temps Δt qui tend vers zéro, et cet intervalle Δt lui-même. On appelle de telles grandeurs des « dérivées par rapport au temps ». On écrit, depuis Leibniz, v = dr/dt. Quant à l'accélération, a = dv/dt =

dérivée d'une dérivée, c'est une dérivée « seconde ». (65-66bc)

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.