Chaotic systems
APRIL 11, 2021 (updated on January 30, 2024)
Table of contents
October 15, 2017, 22:21, Alyssa Milano tweets: “If you have been sexually harassed or assaulted, write ‘Me too’ in response to this tweet. At that moment, the wings of the little Twitter bird are quivering. Wing flapping and wing flapping, a big storm breaks out. A virtual tidal wave that brings to the surface a palpable planetary hunger pangs… Why did that tweet make a difference? Were those the right words? Was it the right time?
The MeToo movement undoubtedly marks a turning point, a “bifurcation” in the relationship between men and women. There will be a before and after MeToo. But how to explain this sudden surge that literally “gave people an idea of the magnitude of the problem” [1] ? Simple chains of cause and effect? Butterfly effect?
In an attempt to bring some order to this case, I invite you to dive with me into chaos theory. “Chaos to bring order? It’s a long shot… ” you think. And yet, I’m off to tell you the great love story of order and chaos… The very great story, in 5 articles!
Chaotic systems, a major discovery
Entering the turbulence zone
At first glance, I grant you that the word “chaos”, and above all the imagination it conveys, are not very attractive. Disorder, confusion, dispersion, inconsistency, mess, catastrophe, tumult… is it really necessary to make a theory out of it?
In fact, in mathematics, chaos theory is much more subtle than the great anything it looks like. Specifically, chaos theory studies non-linear dynamic systems. Translation? Chaos theory studies systems whose behavior changes over time in a way that cannot be predicted.
The apple that falls on Newton’s head follows a perfectly linear and predictable evolution. Classical determinism teaches us that knowing all the parameters of a system allows us to easily anticipate its evolution. That’s the theory, and it works for systems where changing the initial conditions doesn’t make much difference. Thus, whether the apple is half a centimeter or 300 km from its reference point, it still falls, and under the same conditions. Except that within half a centimeter, Newton always gets it on the head!
In practice, it is impossible to know with infinite precision the initial conditions of a system. And when a system is very sensitive to initial conditions, the influence of this lack of information on the prediction of its behavior is significant. The system simply becomes unpredictable in the long run. Such systems are said to be chaotic.
Characteristics of chaotic systems
The example of blood circulation gives a good picture of the situation. Inside the artery, the blood flow is regular. This is called laminar blood flow: the trajectories of particles that are adjacent at one moment remain adjacent at the following moments. But if the blood rushes into an aneurysm [2], it is a different matter. Its flow becomes turbulent and chaotic, its appearance complex and random. However, it is not devoid of coherence and order. We will have the opportunity to come back to this in the next article From Chaos to Interdependence.
To sum up, chaotic systems are:
- closed: they do not exchange energy or information with the outside world.
- dynamic: their structure and their very existence are conditioned by their evolution, even if this evolution follows a non-linear equation.
- deterministic and very sensitive to initial conditions: their future behavior is entirely determined by initial conditions, without the intervention of chance. A minute variation in these conditions, however, has considerable consequences; we speak of deterministic chaos.
- unpredictable beyond a certain time limit, because the initial conditions cannot be known with infinite precision.
Wait a minute, butterfly!
Chaos theory was popularized by the image of the butterfly. The flapping of its small fragile wings represents the sensitivity of chaotic systems to initial conditions. This theory has been demonstrated in a meteorological context, but over time, it has also found other applications, particularly in psychology.
So, is the Metoo effect to women’s indoor weather what the butterfly effect is to the weather outside? Indeed, psychology seems to establish certain parallels between the “internal weather of the psyche” and chaos theory. Clinical psychologist Jérémie Vandervoode explains :
“The [internal weather of the psyche] that produces [certain] phenomena and their brutal oscillation, however, involves so many parameters that it becomes impossible to predict the trajectory of all the components taken individually.”
JÉRÉMIE VANDERVOODE [3]
Let us take a little height and look at the MeToo phenomenon not only on the scale of the individual psyche but on the scale of the collective psyche. To understand how a single tweet could have such a big impact and influence in the world, one must understand that chaos theory does not say “small causes produce big effects”, but “a very large number of small causes produce big effects“. Better still, it is the resonance of very many small causes that ends up tipping the system over and dragging all its dynamics towards another type of behavior. We will come back to the resonance phenomenon throughout this series of articles, and especially in the last: MeToo or the other butterfly effect.
For the moment, I would like to focus on the very foundations of chaos theory, which I find challenging.
Critical questions
This questioning seems to me to be a determining factor in trying to understand the dynamics of chaotic systems, and the MeToo movement in particular. Here I lay the foundations, which will be developed in the following articles, with regard to Nassim Haramein‘s theory of the connected universe.
- Non-linear dynamical systems, which are the object of study in chaos theory, obey the law of causality and determinism. According to the theory of the connected universe, however, determinism is not always at work in the universe. It alternates with indeterminism, so that everything is always being determined. Nassim Haramein shows that the link between the deterministic side and the non-deterministic, and therefore non-predictable, side of the universe is made through fractals which, as we shall see, are closely linked to chaotic systems!
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- The problematic of initial conditions refers to that of isolated systems. For if the system is not isolated from its environment at a given time, when is it considered to be in the desired initial conditions? And what happens to the notion of initial conditions in an infinite universe where everything is connected?
- The future behaviour of dynamic systems is entirely determined by the initial conditions, without the intervention of chance. But how is chance defined?
- If the above questioning takes us beyond the standard frame of reference of chaotic systems, then what is the dynamic that really underlies the butterfly effect?
Let’s go back into the chrysalis and try to figure out how it all started…
A bit of history
Chaos theory has its origin in a simple question: is the solar system stable? This question goes back at least to the time of Isaac Newton (1642 – 1727) when he enunciated his law of universal gravitation [4]. If it allows to calculate and specially to predict celestial movements, it is precise only as long as the studied system has only two objects. Starting from three objects, discrepancies appear between the calculations and what is actually observed. In other words, the system becomes chaotic.
The work of Henri Poincaré
From the notion of trajectory to that of resonance
It was the French mathematician Henri Poincaré (1854-1912) who brought this dynamic to light by studying the movement of three bodies in gravitational interaction: the earth, the sun and the moon. He who was looking for proof of the stability of the solar system showed, conversely, the uncertainty hidden behind the law bequeathed by Newton. Thereby he laid the solid foundation for chaos theory.
Precisely, he has shown that minute uncertainties about the initial state of a system, instead of remaining more or less the same over time, will on the contrary increase. And exponentially. Thus, even the slightest deviation from the initial conditions can have considerable consequences, which are impossible to predict in the long term. The system, although perfectly described by the equations, ends up having unpredictable dynamics.
However, over a relatively short period of time on the galactic scale, and transiently, the orbits of the planets are as regular as the movements of a clock. This is why natural phenomena such as tides or eclipses can be predicted with great accuracy.
The physicist Ilya Prigogine (1917-2003) recalls that Poincaré “introduced the crucial notion of “non-integrable dynamic system” [5] and [6]. “Most dynamic systems are indeed non-integratable because there are resonances between the degrees of freedom of the system. That is, random variables that cannot be determined or fixed by an equation are sensitive to certain frequencies. In fact, resonances make the Newtonian way of thinking obsolete because they invite us to think in terms other than trajectories.
“Non-integrability is due to resonances [7] (…) Therefore they introduce a foreign element to the notion of trajectory.”
The question of chance
Poincaré also addressed the notion of chance. At length, since he devoted a chapter to it in his book Science and Methods. Perhaps this is the essence of his thinking:
“A very small cause, which escapes us, determines a considerable effect that we cannot not see, and then we say that this effect is due to chance (…). It can happen that small differences in the initial conditions generate very large differences in the final phenomena; a small error in the former would produce a huge error in the latter. Prediction becomes impossible and we have the fortuitous phenomenon.”
HENRI POINCARÉ [9]
In short, deterministic chaos juxtaposes two a priori contradictory notions. On the one hand, determinism that leaves no room for chance. On the other, chaos which, being by nature unexpected, is based on chance.
The physicist David Ruelle sums up Poincaré’s thinking well, pointing out that the uncertainty of chaos – the sensitivity of chaotic systems to initial conditions – is a source of chance. In other words, “chance corresponds to an incomplete information…” [10] and [11].
But at the time, what has been retained from Poincaré’s work is that “minute variations in initial conditions lead to unpredictable behaviour”. [12]
Lorenz and the butterfly effect
It is this formulation that the mathematical meteorologist Edward Lorenz (1917-2008) popularized with the metaphor of the butterfly. But how did Lorenz come to imagine that the tiny movement of a butterfly’s wings could cause a storm thousands of miles away? Simply because in the course of his own work he found himself confronted with the importance of the initial conditions, as Poincaré was.
It’s the early 1960s and Lorenz is using a rather rudimentary computer installed at the Massachusetts Institute of Technology. Nothing like the computer hardware of the 1970s, which will allow precise and immediate visualization of the complexity of the solar system, for example.
At the moment, Lorenz is trying to model atmospheric convection [13]. To do so, he writes a simplified system of equation [14]. One day, wishing to resume his work where he had left it, he rounded up the values he had found to a hundredth, then reintroduced them into his equations. Then he realizes that simply rounding the values has the same effect as a tiny variation in the initial conditions: he finds himself face to face with a chaotic system!
In an article [15] from 1963, Lorenz explains the theory of the wing flapping of a butterfly, which, however, does not yet bear this name. It was not until 1972, when he gave a lecture at the American Association for the Advancement of Science. The organizer, meteorologist Philip Merilees, chose the title: “Predictability: Can the flapping of a butterfly’s wings in Brazil cause a tornado in Texas?”. Lorenz discovered the title too late to change it, the expression remained. However, he was very quick to take certain precautions in relation to this formulation.
A necessary perspective perspective
He explains:
“Lest the title of this article, “Can a flutter of butterfly wings in Brazil trigger a tornado in Texas”, cast doubt on my seriousness, let alone an affirmative answer, I will put this question into perspective by making the following two propositions:
- If a single flap of a butterfly’s wings can set off a tornado, then so can all the previous and subsequent flaps of its wings, as well as those of millions of other butterflies, not to mention the activities of countless more powerful creatures, especially of our own species;
- If the flapping of a butterfly’s wings can trigger a tornado, it can also prevent it. If the flapping of a butterfly’s wings influences the formation of a tornado, it is not obvious that the flapping of its wings is the very origin of the tornado and therefore that it has any power over the creation of the tornado or not. ” [16]
So it’s not the flapping of the wings itself that causes the tornado. The formation of a tornado is due to the evolution of the atmosphere, which is sensitive to even minute changes.
Continuing his work, Lorenz also highlighted a remarkable property of chaotic systems… which I invite you to discover in the next article From chaos to interdependence !
Key points
- Chaotic systems are closed, dynamic, deterministic and very sensitive to initial conditions, and therefore unpredictable beyond a certain time limit.
- Chaos theory does not say “small causes produce big effects”, but “the resonance of a very large number of small causes produces big effects”.
- Sensitivity to initial conditions was popularized through the metaphor of the butterfly. But it is not the flapping of the wings per se that causes a tornado: tornadoes are formed by the evolution of the atmosphere, which is sensitive to even minute changes.
Notes et references
[1] Alyssa Milano’s tweet follows one from her friend, Charlotte Clymer : “If all the women who have been sexually harassed or assaulted wrote “me too” as a status, we might give people a sense of the magnitude of the problem.”
Characteristics of chaotic systems
[2] An aneurysm is a dilation of the wall of an artery that causes the creation of a pocket inside which the blood changes its behaviour. Read My Story to understand why I chose this unlikely example.
[3] Vandervoode Jérémie, Les processus dynamiques – la théorie du chaos en psychologie [Dynamic processes – chaos theory in psychology], in Le Journal des psychologues, n°203, December 2010 – January 2011, p.70 (in French)
The work of Henri Poincaré
[4] The universal law of gravitation is a law describing gravitation as a force responsible for the fall of bodies and the movement of celestial bodies, and thus for their trajectory. In general, Newton establishes that all bodies with mass exert an equivalent force on each other that attracts them to each other.
[5] PRIGOGINE Ilya, La fin des certitudes [The end of certainties], Paris, éditions Odile Jacob, 1996, p.44
[6] An integral is used to calculate the area under the curve of any function. It is a continuous (not discrete) sum, as if an infinite number of values were added together. Therefore, a non-integrable system is a system whose evolution cannot be calculated indefinitely by equations describing precise trajectories.
[7] The issue of resonances will be discussed in more detail in the articles Irreversibility, memory and entropy and Gravity, entropy and self-organization.
[8] PRIGOGINE Ilya, La fin des certitudes, op.cit., p.127
[9] POINCARE Henri, Science et Méthode [Science and Method], Paris : Ed. Flammarion, 1908, pp.68-69
[10] RUELLE David, Chaos, imprédictibilité, hasard [Chaos, unpredictability, chance], L’université de tous les savoirs, conférence n°218, août 2000 (in French)
[11] On a more positive note, I would say that the notion of chance is about “I don’t know what I don’t know” (see How do we learn? on this statement). And reminds us of both the importance and the influence of what we don’t know (see also the article Synchronicities, a resonance with the unconscious).
[12] Note that this formulation is already more accurate than “small causes have produced large effects”.
Lorenz and the butterfly effect
[13] Atmospheric convection refers to all internal movements of the Earth’s atmosphere resulting from air instability due to a vertical or horizontal temperature difference.
[14] The appropriate equation system, Navier-Stokes, was too complicated for Lorenz’s computer to solve. He therefore simplified the system to keep only three degrees of freedom.
[15] Lorenz, Edward N. (1963). Deterministic nonperiodic flow, in Journal of the atmospheric sciences, 20(2), pp.130-141.
[16] Lorenz, Edward N., “Can a Flapping Butterfly Wing in Brazil Trigger a Tornado in Texas,” Alloy 22 (1993), 42-45 in: The essence of chaos, The Jessie and John Danz Lecture Series, University of Washington Press (1993).
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