April 11, 2021

The butterfly effect 1/5

Chaotic systems


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.

pomme-newtonThe 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 Jeremie 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.  »


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.

  1. fractales 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 !

  2. 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 ?

  3. The future beha­viour of dyna­mic sys­tems is enti­re­ly deter­mi­ned by the ini­tial condi­tions, without the inter­ven­tion of chance. But how is chance defined ?
  1. 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. » [8]


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 :

poincare« 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 ».


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 !

effet-papillonIn 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

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 ;
  • f 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 and 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. A non-integrable sys­tem is the­re­fore 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
[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 Chance or Synchronicity ?).
[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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