May 5, 2021
The butterfly effect 2/5
From chaos to interdependence
« 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 ?
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)
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…
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…
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 spacetime 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 spacetime, 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 spatiotemporal 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 !

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 selforganization.
[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) Spacetime as Information — An Ordering Principle of Living Systems