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May 5, 2021

The butterfly effect 2/5

From chaos to interdependence

attracteur-de-lorenz-chaos

« In eve­ry chaos, there is a cos­mos, in eve­ry disor­der, a secret order. » [1]


This article fol­lows the one on chao­tic sys­tems, in which I recal­led the deci­sive role of Henri Poincaré and Edward Lorenz in the dis­co­ve­ry and foun­da­tion of chaos theory.

Lorenz also high­ligh­ted a fas­ci­na­ting pro­per­ty of chao­tic sys­tems : in the long run, they end up oscil­la­ting around what appears to be a finite num­ber of values. It is said that these values accu­mu­late on an attrac­tor.

If attrac­tors seem then inse­pa­rable from chaos, they could well be part of ano­ther order of things, high­ligh­ted by Nassim Haramein’s theo­ry of the connec­ted uni­verse. This is what I invite you to dis­co­ver in this article !

                 

Strange attractors

Chaotic sys­tems seem to evolve in a para­doxi­cal way. On the one hand, their sen­si­ti­vi­ty to ini­tial condi­tions makes them unpre­dic­table over time. On the other hand, howe­ver, they end up repro­du­cing the same pat­terns. In the end, if you look at things over the long term, they are rela­ti­ve­ly pre­dic­table and… insen­si­tive to star­ting condi­tions ! In order to get an idea of what will emerge from the chaos, we must then look at the fixed and per­io­dic points, the famous attrac­tors towards which they end up converging.

The attrac­tor reflects the move­ment of the sys­tem, it makes it pos­sible to rea­lize its speed and posi­tion. To a regu­lar sys­tem will be lin­ked a very simple attrac­tor, such as a circle or an ellipse. But for a chao­tic sys­tem with three variables, the attrac­tor becomes more complex

This can be seen by mode­ling what is cal­led phase space. This is the space where the dif­ferent phases through which the sys­tem passes are loca­ted. A phase change being cha­rac­te­ri­zed in phy­sics by a sud­den change in the state of the sys­tem. If you’ve ever made home­made mayon­naise, you know what I’m tal­king about. Indeed, you observe the mixing of egg yolk, mus­tard, vine­gar and oil taking place gra­dual­ly, but irre­ver­si­bly, up to a cer­tain point. Beyond this point, the sys­tem toggles and changes phase : the « mayon­naise » state appears.

If we model the phase space of a chao­tic sys­tem, we obtain a rather sin­gu­lar « curve » [2] : it repro­duces the same move­ment all the time, without ever over­lap­ping. For example, Lorenz’s attrac­tor (see the main illus­tra­tion in this article) consists of two « loops » that look like a… but­ter­fly ! In chaos theo­ry, we call these strange attractors.

               

Strange attractors, really ?

diagramme-bifurcation-chaos

Bifurcation dia­gram to chaos by per­iod dou­bling [3]


Although strange attrac­tors can take many forms, they have one thing in com­mon : their struc­ture repeats itself iden­ti­cal­ly, ad infi­ni­tum. It’s a frac­tal. This struc­ture can be seen in the dia­gram above.

There is one thing, howe­ver, that can­not be seen in the illus­tra­tion of the Lorenz attrac­tor, and that is that their values do not exist on a single « sur­face » but in a depth of « surfaces ».

The fact remains that even if a chao­tic sys­tem evolves towards the attrac­tor in an erra­tic and unpre­dic­table way, soo­ner or later it will converge towards it. It will thus evolve from appa­rent chaos to a cer­tain regu­la­ri­ty. In the case of the Lorenz attrac­tion, howe­ver, the num­ber of laps over one region or ano­ther remains dif­fi­cult to pre­dict. But wha­te­ver the star­ting point — and the­re­fore wha­te­ver the ini­tial condi­tions — all tra­jec­to­ries will even­tual­ly pass through one or the other region, and with the same fre­quen­cy.


« Over the years, tiny dis­tur­bances don’t increase or decrease the fre­quen­cy of wea­ther events like tor­na­does, the most they can do is change the order in which they occur. »
[4]

 

So even if Mr. Butterfly didn’t flap its wings, Mrs. Tornado would still show up. It would just come at a dif­ferent time. In other words, by varying the star­ting condi­tions slight­ly, you get — or don’t get — a tor­na­do at a given time in a given place. However, in the end, both evo­lu­tions (the two loops of the attrac­tor) will contain as many tor­na­does as each other. Strange attrac­tors are said to reveal a conti­nuous spec­trum of fre­quen­cies.

Turbulence phe­no­me­na, stu­died by fluid mecha­nics, are not limi­ted to meteo­ro­lo­gy. They are also obser­ved in medi­cine, with the beha­vior of blood in an aneu­rysm [5].

                  

The example of the aneurysm

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. The only ener­gy losses are rela­ted to vis­co­si­ty, which pro­vides uni­form resis­tance to blood flow. However, when blood rushes into an aneu­rysm, chaos seems to set in : the blood flow becomes tur­bu­lent.

The condi­tions that deter­mine whe­ther a flow is lami­nar or tur­bu­lent are given by the Reynolds num­ber (Re). It is a ratio bet­ween the iner­tial forces rela­ted to the flow velo­ci­ty and the fric­tio­nal forces rela­ted to the viscosity.

Re = 2pvr/ η (where p is den­si­ty, v is velo­ci­ty, r is radius and η is viscosity)

anevrisme-structure-dissipative

Above a cri­ti­cal speed (high Re), the sys­tem reaches a bifur­ca­tion point. The flow becomes tur­bu­lent, then the blood flow gives an impres­sion of disor­der and com­plexi­ty. It is in fact very struc­tu­red and consists of « whirl­pools ». Although the nature of the blood sys­tem remains the same, its macro­sco­pic struc­ture changes.

Like the dyna­mics of chao­tic sys­tems, vor­tex dyna­mics fol­low a frac­tal geo­me­try. The divi­sion of large swirls into smal­ler ones allows a trans­fer of ener­gy from large to small scales. We talk about ener­gy cas­cades that cause a strong dis­si­pa­tion of ener­gy [6].

The dyna­mics of tur­bu­lence phe­no­me­na are iden­ti­cal wha­te­ver the scale consi­de­red : aneu­rysms, meteo­ro­lo­gi­cal swirls as we have seen, but also large struc­tures of the uni­verse such as clus­ters of galaxies. And a lit­tle tour of the connec­ted uni­verse will help us unders­tand why this is so !

                      

Chaotic Systems in the Connected Universe

A farewell to isolated systems

If sen­si­ti­vi­ty to ini­tial condi­tions is ques­tio­nable in chao­tic sys­tems, it is no lon­ger rele­vant in the connec­ted universe.

The key is to unders­tand that, first, no sys­tem is or becomes chao­tic. All sys­tems are com­plex [7] by nature. However, they may appear deter­mi­nis­tic over a per­iod of time and at the time they are stu­died. Isolated and then pro­vi­ded with ini­tial condi­tions, they some­times expe­rience a tran­sient stea­dy state. It is this state that makes it pos­sible to pre­dict eclipses, for example.

Secondly, the pro­blem of ini­tial condi­tions refers to that of iso­la­ted sys­tems. Indeed, if a sys­tem is not iso­la­ted from its envi­ron­ment at a given time, when is it consi­de­red to be in the desi­red ini­tial conditions ?

In the connec­ted uni­verse, it makes no sense to talk about linear sys­tems, chao­tic sys­tems or iso­la­ted com­plex sys­tems — and the­re­fore about ini­tial condi­tions. There is only one com­plex sys­tem whose variables are in constant inter­ac­tion : the uni­verse itself. It is com­po­sed of com­plex sub­sys­tems, which are lin­ked and inter­act with each other through feed­back. They fol­low frac­tal dyna­mics, where each level contains more infor­ma­tion than the pre­vious one.

In such a uni­verse, is it so weird that attrac­tors are strange ?

A frac­tal uni­verse also implies that deter­mi­nism and inde­ter­mi­nism coexist, so that eve­ry­thing is always being deter­mi­ned. This leads to the for­ma­tion of struc­tures constant­ly on the bor­der­line bet­ween order and chaos :


« This means that [the beha­viour of these struc­tures] is a subtle balance bet­ween the need for order so that [they] do not dis­solve and the need for free­dom to allow them to evolve, trans­form, adapt. »
[8]

                

Dynamic conditions Vs initial conditions…

resonance-frequences-chaos

A com­plex sys­tem does not change pro­por­tio­nal­ly to the change in its para­me­ters. On the contra­ry, he sees the effect cau­sed by a change, howe­ver small, beco­ming dis­pro­por­tio­nate. As a result, its beha­viour and evo­lu­tion are impos­sible to pre­dict. However, it is not the num­ber of sys­tem para­me­ters that is an impe­di­ment to pre­dic­tion. It is the influence that these para­me­ters exert on each other through the feed­back loops that link them toge­ther. It’s the num­ber of feed­backs [9] that counts. And it is not feed­back from one ele­ment to ano­ther, but from all ele­ments to all others simul­ta­neous­ly and conti­nuous­ly. Like an ocean in per­pe­tual motion, from which one can­not iso­late, if not arbi­tra­ri­ly, the sligh­test transformation.

Everything is entan­gled [10] in com­plex sys­tems, which are, the­re­fore, irre­du­cible to their ele­men­ta­ry components.

« Each com­ponent contri­butes to glo­bal beha­vior through its local inter­ac­tions with others. Isolating pieces of the sys­tem radi­cal­ly changes the beha­vior of the whole. The clas­si­cal ana­ly­ti­cal method, which consists in cut­ting a com­pli­ca­ted set into sup­po­sed­ly sim­pler sub­sys­tems to stu­dy their beha­vior and trying to recons­truct the ove­rall beha­vior by com­bi­na­tion, fails. (…) A com­plex sys­tem can only be stu­died glo­bal­ly. » [11]


It is pos­sible to believe that a sys­tem is linear in nature. But when we rea­lize that it is the defi­ni­tion of arbi­tra­ry ini­tial condi­tions that iso­lates the sys­tem, it is linear only as long as it is lin­ked to these condi­tions. Defining ini­tial condi­tions masks the dyna­mics real­ly at work in a uni­verse where nothing is iso­la­ted from nothing. This makes us focus on the chaos gene­ra­ted by the lack of know­ledge of the ini­tial condi­tions as a whole. But is this lack of infor­ma­tion real­ly detrimental ?

                 

… in all systems…

particules-quantiques

If you are fami­liar with the theo­ry of the connec­ted uni­verse, it is as if we don’t take into account the infor­ma­tion enco­ded until this point on the space-time frame.

Rather, we set an arbi­tra­ry time t. And we leave aside the fact that the ini­tial condi­tions as defi­ned at this point depend on the pre­vious condi­tions that brought the sys­tem to this point. This ignores the dyna­mics of the sys­tem. Even so, it high­lights a frac­tal geo­me­try as shown in the graph of the bifur­ca­tion to chaos pre­sen­ted above. And that, the­re­fore, wha­te­ver the ini­tial condi­tions cho­sen, this does not change the gene­ral beha­vior of the system.

Considering that there are stable sys­tems and uns­table sys­tems is mis­lea­ding. There are only com­plex sys­tems with stable and uns­table phases. The linear pro­gres­sion of cer­tain variables can be fol­lo­wed during the stable phase. One can dedu­ced from this that any exter­nal dis­tur­bance will be dam­ped by the sys­tem and will not fun­da­men­tal­ly change the tra­jec­to­ry of the variables that make it up. However, this does not mean that the sys­tem is eter­nal­ly stable.

It is not because sys­tems are stable that they dam­pen « exter­nal » dis­tur­bances  [12], it is because the accu­mu­la­tion of ener­gy brought to the com­plex sys­tem through dis­tur­bances is not suf­fi­cient to make the sys­tem move. In other words, there are no stable sys­tems that dam­pen dis­tur­bances, there are only dis­tur­bances that are not yet nume­rous enough to cause sys­tems in stable phase to fall into insta­bi­li­ty [13].

                     

… on all scales

In his confe­rence « Chaos, unpre­dic­ta­bi­li­ty, chance », phy­si­cist and mathe­ma­ti­cian David Ruelle reminds us that quan­tum mecha­nics neces­sa­ri­ly calls for chance. According to him, the lat­ter cor­res­ponds to incom­plete infor­ma­tion. In res­ponse to the ques­tion whe­ther quan­tum mecha­nics should not be used in the dis­cus­sion of the rela­tion­ship bet­ween chance and deter­mi­nism, he poin­ted out that quan­tum effects appear to be negli­gible, espe­cial­ly for the move­ment of the stars.


« For a given class of phe­no­me­na, seve­ral theo­ries are in prin­ciple appli­cable and we can choose the one we want ; for any rea­so­nable ques­tion, the ans­wer should be the same accor­ding to whe­ther one takes one theo­ry or ano­ther ; in a field of appli­ca­tion that is valid for both, one should have the same ans­wer. So in prac­tice we will use the theo­ry that is easiest to apply, in the cases that inter­est us — dyna­mics of the atmos­phere or move­ment of the pla­nets — it is natu­ral to use a clas­si­cal theo­ry and not to try to do quan­tum mecha­nics. After that, there will always be time to check that the quan­tum or rela­ti­vis­tic effects that were neglec­ted were real­ly negli­gible and that, all in all, all the ques­tions that were asked were rea­so­nable ques­tions. » [14]


Quantum mecha­nics as it is cur­rent­ly pre­sen­ted — i.e. the quan­tum world as it is cur­rent­ly inter­pre­ted — shows its limits as soon as a link with cos­mo­lo­gi­cal phy­sics can­not be esta­bli­shed. So per­haps the ques­tion is not whe­ther quan­tum mecha­nics should be taken into account, but rather the rela­tion­ship bet­ween the quan­tum world and the cos­mo­lo­gi­cal scale.

                   

Precision or prediction ?

This is what Nassim Haramein does in his uni­fied field theo­ry. His approach raises a ques­tion : what hap­pens to the ini­tial condi­tions in an infi­nite universe where eve­ry­thing is connec­ted at all scales ? A ques­tion that leads to ano­ther : isn’t space-time, and ulti­ma­te­ly mat­ter, taking their source in the infi­ni­te­ly small, the place where the ini­tial condi­tions also take theirs ?


« Even in [quan­tum theo­ry], it is reco­gni­zed that an infi­nite mea­su­ring device could veri­fy with com­plete deter­mi­nis­tic cer­tain­ty the quan­tum state and spatio-temporal coor­di­na­tion of all par­ticles in a given space. Alas, such an ins­tru­ment can­not exist because it would col­lapse into a black hole. Yet there is a black hole of this magni­tude that mea­sures the state of all fun­da­men­tal quan­ta at any given time : it is the uni­verse (…). » [15] and [16]

 

Chaos theo­ry says that you can pre­dict the pre­sence of a hur­ri­cane in a given place at a given time, pro­vi­ded you know with extreme pre­ci­sion the ini­tial condi­tions, i.e. the air move­ments down to the sligh­test flap of a but­ter­fly’s wings. In prac­tice, and regard­less of the theo­ry used, kno­wing the ini­tial condi­tions in order to make this pre­dic­tion is impossible.

But with the connec­ted uni­verse theo­ry, it becomes per­fect­ly secon­da­ry whe­ther they are known or not. Knowing the dyna­mics of the uni­verse is enough to pre­dict that if one suf­fi­cient­ly feeds the sys­tem with a cer­tain inten­tion, it will have no choice but to mani­fest it.

In the next article Irreversibility, memo­ry and entro­py, I conti­nue to explore this theo­ry from other angles. Stay tuned !
          

     


Key points

  • The struc­ture of the strange attrac­tors is repea­ted iden­ti­cal­ly, ad infi­ni­tum : it is a fractal.

  • There are no chao­tic sys­tems, all sys­tems are com­plex in nature. The uni­verse itself is a com­plex sys­tem, com­po­sed of com­plex sub­sys­tems, whose variables are in constant interaction.

  • In a com­plex sys­tem, it is the num­ber of feed­backs from all sys­tem para­me­ters to all others, simul­ta­neous­ly and conti­nuous­ly, that counts.

  • There are nei­ther stable nor uns­table sys­tems. There are only dis­tur­bances that are not yet nume­rous enough to tip sys­tems in stable phase into instability.

  • Knowing the dyna­mics of the uni­verse rather than the ini­tial condi­tions is enough to pre­dict that if one suf­fi­cient­ly feeds the sys­tem with a cer­tain inten­tion, it will have no choice but to mani­fest it.     
                        

                      

                         

                          



Notes and references



Strange Attractors

[1] JUNG Carl Gustav, The Archetypes and the Collective Unconscious, Princeton University Press, 1934, p.32
[2] In rea­li­ty, it is not real­ly a curve, or even a sur­face, but an orde­red set of dis­crete values (i.e. a set contai­ning a finite num­ber of values bet­ween any two values) consti­tu­ted by the « chao­tic » dyna­mics of the sys­tem.
[3] Source : Wikipedia
[4] 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.

   

The example of the aneurysm

[5] An aneu­rysm is a dila­tion of the wall of an arte­ry that causes the crea­tion of a pocket inside which the blood changes its beha­vior. Read My Story to unders­tand why I chose this unli­ke­ly example.
[6] See also the article about entro­py.

 

Chaotic Systems in the Connected Universe

[7] A com­plex sys­tem is a set made up of a large num­ber of inter­ac­ting enti­ties that prevent the obser­ver from pre­dic­ting its feed­back, beha­viour or evo­lu­tion through com­pu­ta­tion. According to Wikipedia, Complex System.
[8] ZWIRN Hervé, Les sys­tèmes com­plexes  [com­plex sys­tems], Paris, Editions Odile Jacob, 2006, p.10
[9] The notion of feed­back will also be addres­sed in article 4 Gravity, entro­py and self-organization.
[10] Two entan­gled par­ticles can­not be consi­de­red as inde­pendent, regard­less of the dis­tance bet­ween them. They form a unique sys­tem where an action on one has an ins­tant impact on the other. In the connec­ted uni­verse, it’s all par­ticles that are entan­gled. To learn more about the entan­gle­ment phe­no­me­non, you can read the article Indeterminism and entan­gle­ment.
[11] ZWIRN Hervé, Les sys­tèmes com­plexes, op.cit., p.19
[12]  In any case, there are no « exter­nal » dis­tur­bances in a uni­verse where all sys­tems are inter­de­pendent : nothing is exter­nal to any­thing.
[13] We will have the oppor­tu­ni­ty to come back to this in article 5 MeToo, or the other but­ter­fly effect.
[14] Ruelle David, Chaos, impré­dic­ti­bi­li­té, hasard [Chaos, unpre­dic­ta­bi­li­ty, chance]

, L’université de tous les savoirs, confé­rence n°218, août 2000
[15] « As soon as we consi­der that the mass of our obser­vable uni­verse is contai­ned within its cur­rent­ly mea­su­red radius, our uni­verse thus obeying the Schwarzschild condi­tion or black hole condi­tion ».
[16] BROWN William and HARAMEIN Nassim (2014, 23 January) Space-time as Information — An Ordering Principle of Living Systems

 




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