Category Archives: Chaos

From Creation to Creativity…


Everything in the Universe is fundamentally a system of some sort.  And all systems fundamentally have the same Universal Dynamics…

Matrix of Universal Dynamics - Copyright - Kieran D. Kelly


Our Universe is fundamentally the interplay of 2 things; Energy and Probability.  Excessive Energy can cause Incompressible Dynamics.  Probability can cause Random Energy Clustering which can ultimately lead to Natural Reinforcement.  And the interplay of both incompressible dynamics and probabilistic reinforcement can cause the emergence of Self-Integrated Matter.

Matrix of Energy Dynamics - Copyright - Kieran D. Kelly


Everything in the universe is ultimately some type of “System”.  These systems have a range of behavior from Simple Non-Adaptive Systems to Complex Adaptive Systems.  All systems are essentially the interplay of 2 things; Energy and Feedback.   Excessive Energy can cause Incompressible Dynamics.  Feedback can cause Natural Reinforcement.  And the interplay of Incompressible Energy and Natural Reinforcement can cause of the emergence of Self-Integrated Complexity.

Matrix of System Dynamics - Copyright - Kieran D. Kelly


The Brain is a Complex Adaptive System.  The Brain is capable of modeling Data from the External World.  The more data the brain is exposed to the more likely there will be diversity within the data.  The Brain is dominated by 2 things; External Data and Conscious Evaluation.  An excess of data can produce a great diversity of data.  Experimentation can extract information and knowledge from data.  And the interplay of both a Diversity of Data and Conscious Evaluation can generate Spontaneous Self-Integration and Deep Intuition… Matrix of Cognitive Dynamics - Copyright - Kieran D. Kelly


The Mind is a Complex Adaptive System.  The Mind is capable of modeling Thoughts and Intuitions from its own Internal World.  The more thoughts the mind is willing to entertain the more likely there is diversity within the thoughts.  The Mind is dominated by 2 things; Thoughts and Feelings.  A lot of thoughts can produce a great diversity of intuition.  Thinking can extract ideas from thoughts and intuitions.  And the interplay of both a Diversity of Intuition and Emotional Reinforcement can generate Spontaneous Self-Integration and Deep Creativity…

Matrix of Creative Dynamics - Copyright - Kieran D. Kelly

What’s Driving Evolution?

Wrdcloud of what Drives Evolution


According to the Second Law of Thermodynamics we live in a universe that irreversibly decays over time.  But if this is indeed the case, then it begs the question:   How does Evolution’s Spontaneous and Progressive Complexity occur without some form of External Organizing Force?”


Evolution vs. Entropy

The Second Law of Thermodynamics (SLOT) is a law of physics that deals with “Spontaneous Change” (i.e. change that occurs without any external direction, change that happens all by itself…)

The SLOT is, more precisely, the law of physics that deals with how energy distributes itself within a thermal system, always moving spontaneously and irreversibly to “Thermal Equilibrium”.

In everyday terms the SLOT is simply the fact that hot coffee and cold milk, if left unstirred, will spontaneously mix themselves (in both composition and temperature), and will never spontaneously un-mix.

Despite the fact that this seems rather obvious and trivial behavior, the SLOT is nonetheless considered to be one of the most fundamental and important laws of physics — and the reason for this exalted status is that the SLOT is both a “Probabilistic Law” and also the “Law of Maximum Entropy”!

“Entropy” is a concept that deals with amount of “disorder” in a system, and it is widely understood that the spontaneous gravitational pull to maximum entropy is not restricted to simple thermal systems; but that all systems, if left undisturbed, will spontaneously gravitate towards a state of maximum disorder — a state that would seem to be the exact opposite of Nature’s “spontaneously self-organized complexity”.

This apparent conflict between physics and natural evolution obviously begs the question:   “How does Evolution manage to spontaneously generate such incredible Complexity in the face of the SLOT?”

How can natural complexity spontaneously arise in a universe dominated by the SLOT and its spontaneous and irreversible pull to disorder?  What exactly is the “Source” of all of Nature’s spontaneous order and complexity?

The Export of Entropy

In 1977, the Belgian chemist Ilya Prigogine won the Nobel Prize for Chemistry, for his work on his “Theory of Dissipative Structures”.  Prigogine’s theory suggests that complex ordered systems can indeed come into existence if these systems are open and capable of “exporting”  their internal disorder, to the external environment.

But while this theory would seem to go some way towards solving the paradox of how order can occur without negating the SLOT, it still does not manage to identify what fundamental forces are actually driving evolution to evermore progressive complexity.   Physics has as yet offered no explanation for “evolution’s progressive arrow of time”

As it turns out however, the resolution of this paradox is actually quite easy.  To resolve this apparent conflict  between physics and natural evolution we need merely to focus on a very simple fact that has been consistently overlooked about the “probabilistic” SLOT; the fact that it relies heavily on the “Law of Large Numbers (LLN)”…

The LLN

Most people are familiar with the concept that if we toss a coin four times, we won’t necessarily get a 50/50 split of heads and tails: indeed, we could actually get 4 tails in a row.  But if we toss the same coin a million times, we will almost certainly get something close to a 50/50 split.  It is the LLN that ensures that one million coins tosses will produce an average of 50% heads and 50% tails.

[Note: In the simplest possible mathematical terms, the reason the LLN works so well is that the number of independent tosses (i.e. 1,000,000) is significantly larger than the number of options available to each toss (i.e. 2 – heads or tails).]

The SLOT states that left undisturbed all systems gravitate towards the “most probable state”, a state that is referred to as “thermal equilibrium”.  In reality however, the achievement (and sustainment) of thermal equilibrium relies heavily on the number of independent elements (of the system) being significantly larger than the number of energy options available to each element.  Which means that the chances of any “statistical deviations” from the “most probable state” are extremely small, and consequently the system as a whole will (virtually) always exhibit uniformity.

So although on the “microscopic level” (of particle interaction) there is a lot of energetic dynamics and non-equilibrium abnormalities, these dynamics and abnormalities are normally invisible on the macro “system level” thanks to both the “Damping” and “Balancing Effects” of the LLN…

The RLLN

Our universe is fundamentally a universe of “systems”, and the probabilistic pull of equilibrium is a concept that is applicable to all fluid and fluid-like systems.

Now, in a thermal system there are billions of tiny particles which interact through collisions, but other than that we can more or less say that they behave completely independently of each other.

Systems however, where the parts – be they particles, elements, components, entities, agents, organizations, etc – behave independently of each other are actually quite rare.  Many systems are populated by adaptive elements or agents, and the behavior of these agents has a tendency to weaken the gravitational pull of equilibrium by engineering the “Reverse Law of Large Numbers (RLLN)”…

[Note: Since the LLN relies on the number of independent elements being significantly larger than the number of options available to each element, there are therefore two things can engineer the RLLN and they are:  either the number of independent elements in the system comes down, or, the number of options available to each element goes up…]

RLLN 1:  Emergent Positive Feedback

In all fluid-like systems, the LLN ensures the spontaneous movement to a “global equilibrium”; however for very small regions within these systems, there are not enough particles to ensure a “local equilibrium”.  At the very lowest level within all systems, random fluctuations are undampable and occurring all the time which means that local imbalances are constantly, and randomly, flittering in and out of existence.

Occasionally these random temporary fluctuations can randomly be very persistent.  In a thermal system this is naught but a mere statistical curiosity, but in a complex adaptive systems it can easily happen that some parts within the system will begin to adapt to these persistent fluctuations; and often such adaptation can serve to amplify the imbalance even further, and in so doing, further extend the fluctuation’s duration.  Thus random local fluctuations can lead to the localized emergence of positive feedback which reduces the independence of the elements and ultimately has an unbalancing and reversing effect on the LLN.

RLLN 2:  Insufficient Negative Feedback

Positive Feedback however is not the only thing that can engineer the RLLN.  Since the LLN effectively operates like a negative feedback system (in that it dampens a system to a equilibrium) it should be no surprise that the movement away from equilibrium could also be the result of insufficient negative feedback.

So although complex fluid-like systems might gravitate towards equilibrium, many can hold themselves some distance away from equilibrium by exhibiting excessive undampable adaptation and innovation.  Adaptation and innovation effectively increases element “Optionality” and such increased optionality among the elements of the system can also engineer the RLLN…

Self-Integration For Free

So the reality of probability driven dynamics in the natural world is that just as the LLN pulls a system to thermal equilibrium, so too the RLLN can hold, or drive, a system away from equilibrium.

But ultimately what is most interesting about all of this probabilistic behavior is that: while strong positive feedback in isolation can cause the emergence of self-reinforcing local segregation; and while insufficient negative feedback in isolation can cause the surfacing of incompressible innovative diversity; the most interesting stuff actually occurs at the intersection between the two…

Positive reinforcement in a system of great diversity can spontaneously produce surprisingly complex “Integrated Diversity”.  So in other words,

with the co-emergence of diversity

Complex-Integration comes for Free!..


Natural Complexity

Evolution’s progressive complexity is often portrayed as spontaneous “Self-Organization”, but this is not the exactly accurate.  The secret sauce of evolution’s spontaneous and progressive complexity is actually spontaneous “Self-Integration”.

In the simplest possible terms, Natural Complexity emerges from the finely-tuned self-integration of co-emergent self-organized diversity; and as a consequence “the complex whole is forever becoming greater that its less complex parts”…

Matrix of System Dynamics - Copyright - Kieran D. Kelly

So there we go, Natural Complexity explained (by mathematical probability).  “Easy Peasy Lemon Squeezy”…

In a universe supposedly dominated by the SLOT what drives nature’s progressive evolution is simply the mathematical interplay of the two distinct forms of the Reverse Law of Large Numbers…

What drives evolution’s spontaneous and progressive complexity is the interplay of insufficient negative feedback and strong positive feedback; or in other words what drives evolution is The Interplay of Random Innovation and Natural Reinforcement…

Matrix of Evolution Dynamics - Copyright - Kieran D. Kelly

Coarse Synchronicity


Lorenz Attractor

Chaos & Complexity are related; both are forms of “Coarse Damping”.  While complexity is a form of coarse damping in “structure”, Chaos is a form of coarse damping in “Time”.

Chaos is Coarse Synchronicity.


PART 1 – EMERGENT BEHAVIOR

Fine vs. Coarse

Imagine you are using a camera with a large telephoto lens and you are trying to focus in on a subject in the distance.  As you focus, you adjust the lens back and forth, clockwise and anticlockwise.  Now imagine the lens does not move smoothly clockwise and anticlockwise, but instead has a limited number of preset positions (say 5 to the left, 1 midway, and 5 to the right).  If this were the case you would find it impossible to truly focus on your subject (unless the focus length just happened to be exactly at one of the 11 preset positions), the best you could do is jump back and forth between the 2 closest positions.

Furthermore the degree of focus depends on the step-size between the 2 closest positions; the larger the step size, the less focused the shot.  In this focusing set-up, it is the number of preset positions available that determines the step-size between each individual position; and the smaller the number of pre-set positions, the larger the step-size becomes.

Imagine the volume control on your home music system.  A smooth continuous dial effectively offers an infinite number of positions up to maximum volume, and so the step-size is in effect zero.   A dial with 11 preset levels (from 0 to 10) is obviously coarser in terms of step-size but nonetheless still usable to achieve the type of volume you want.  It becomes more difficult if the dial’s presets are just “Off”, “Low”, “Medium”, and “High”.  And you have no real control, if the dial basically allows you only “Off” or “On”; your step-size is now a single jump from zero to maximum volume…

In both these systems the step-size determines the coarseness of calibration.  This concept of degrees of fine or coarse calibration can be applied to the behavior of all systems, of any type.  Imagine if your personality only had 2 modes of expression, super nice or super angry, I suspect the vast majority of people would think your behavior a bit unpredictable.

The Butterfly Effect 002

The Butterfly Effect

The common perception about chaos is that it is all about unpredictability.  This unpredictability of chaos is popularly known as “The Butterfly Effect”, a sort of exaggerated version of the domino effect; a small change over here can cause a large change over there sometime down the line.  The scientific phraseology is that systems which exhibit chaos are “Sensitive Dependent on Initial Conditions” (SDIC).

The whole initial conditions thing relates to the classical physics idea that if we know all the forces currently acting on a system then the only remaining thing that we need to know in order to predict the future behavior of the system, is the exact state of the system as it is right now.  If a system is overly sensitive to these “initial conditions” it makes prediction impossible because we need to know these “initial conditions” with an immeasurable degree of exactness.

This explanation of chaos however is somewhat misleading.  While not untrue, this explanation places too much emphasis on unpredictability and fails to capture the reason for the extreme sensitivity.  SDIC is the result of internal interactions being too “coarse” to negotiate stability, which leads instead to the emergence of tipping points. 

The unpredictability of chaos might be better described as “Sensitive Dependence to Emergent Tipping Points”

Emergence of Tipping Points

Now imagine we have or we design some sort of self-stabilizing or self-balancing system.  Such a system would require the ability to find its way to the exact point of balance; this point being the “true-equilibrium”.

Many such self-stabilizing systems exist in nature; most of which self-stabilize in the most obvious way; that is they start off coarse-tuning and subsequently fine-tune their way to equilibrium.  If the system however is unable, or becomes unable, to reduce coarse-tuning to fine-tuning, it will be unable to micro-adjust and hone-in on the single true-equilibrium.

When the step-size of the internal adjustments remains too coarse, or become too coarse, it causes constant overshooting of the true-equilibrium, which means that the unobtainable equilibrium has now become a point about which the system oscillates.  [Think of the classic so-called business cycle of economic growth and recession]. 

At first glance it might appear that all such oscillations look the same, but actually they come in two different flavors.  The emergence of an unobtainable equilibrium means that there are now two different path trajectories that the system can take.  The oscillation can be a “tick-tock” oscillation or a “tock-tick” oscillation.  Depending on which side of the equilibrium the system evolves to, determines which path trajectory the system will take.  So although the system cannot find true-equilibrium, it nevertheless still exists, only now it behaves as a tipping point.

TwoLorenzOrbits

Many Evolutionary Paths

The inability to obtain equilibrium means, the behavior of the system has become sensitive to the emergence of a tipping point.  Given any two different sets of initial conditions we now find that their future evolutions are either in phase with each other, or out of phase with each other.  The coarseness of the step-size has thus caused two different (but complimentary) evolutionary paths to emerge…

But it doesn’t stop there!  The coarseness of the step-size not only determines the ability to self-synchronize to equilibrium, it also can determine whether the system is even able to synchronize the two legs of the back and forth oscillation.

If we were able to manually adjust the coarseness of the internal adjustment we would find that as we increase the coarseness of the step-size we make it increasing difficult for the system to synchronize its own internal self-balancing.  The inability to synchronize a two-step balancing leads to a four-step balancing; the inability to synchronize a four-step balancing leads to an eight-step balancing; and so on to infinity…

This process of continual bifurcation is known as the “period doubling route to chaos”.  Each bifurcation results in doubling the number of internal tipping points, which ultimately causes an infinite number of different evolutionary paths to emerge…

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[Many mathematical models exhibit this coarse behavior, the best known of which is the logistic map of population growth.  So for those of you more mathematically inclined, click here for a description/analysis of The Behavior of the Logistic Map.]

Emergence of Desynchronized Diversity


PART 2 – COARSE EQUILIBRIUMS & STRANGE ATTRACTORS

Emergence and Unpredictability

The road to unpredictable chaos may be a gradual breakdown of synchronicity in behavior – leading to an infinite number of different evolutionary paths – but what emerges from this breakdown however is still an equilibrium of sorts.  Each evolutionary path is merely a different version of a previously hidden but now emergent “coarse equilibrium”.

This emergent structure – which results from ever increasing coarse synchronicity – is known in chaos as an “attractor”.  In actuality true-equilibrium is itself also an attractor, just not a very interesting attractor.  True-equilibrium is a “point attractor” which always pulls the system to the same place, regardless of where it starts from.  An emergent attractor however has a diversity of evolutionary paths that it is capable of pulling a system to; and the more emergent the attractor the greater the diversity of the evolutionary paths.

Chaotic Behavior of “Chaos” has what is known as a “strange attractor” – because of its infinite number of evolutionary paths.  This infinite number of paths however is not what makes the system unpredictable.  What makes the system unpredictable is the sheer density of emergent tipping points that system encounters; each one of which alters the behavior of the system, and any one of which can significantly impact its future evolution.  Consequently the more tipping points there are, the stranger the attractor, and the more unpredictable the system’s behavior becomes.

Coarse Deformation

Chaos Theory has revealed the existence of hidden attractors.  Attractors act on a system like a gravitational restraining force or an elastic restoring force.  What chaos theory shows us is that the type of restraining force (of equilibrium attractor) that acts on a system ultimately is determined by the degree of “coarseness of internal interactions”.

Finely-balanced/short-range interactions lead to highly-symmetric uninteresting conforming behavior; coarsely-balanced/long-range interactions on the other hand lead to an interesting mix of creative non-conforming asymmetric behavior.  It is as if coarse damping deforms the true-equilibrium behavior in a manner analogous to how excessive stress causes the deformation of elastic materials.  It is as if increasing the coarseness of step-size causes the emergence of higher dimensional attractors and the associated diversity of surprising patterns of behavior.

Strange Attractor-4


PART 3 – CONCLUSION

So Chaos, it turns out, is actually a form of Coarse Damping to Equilibrium. 

Chaos is the Coarse Synchronization of coarse competing forces resulting in a coarsely synchronized equilibrium…

So why is this important? It is important because Chaos is showing us that there is more to equilibrium than a finely-tuned, highly symmetric, but fundamentally bland and featureless uniformity.

Practically everything in the universe is both some form of system in itself, and part of a larger system in some form or other; and equilibriums are central to systems.  Chaos is the study of what can emerge from coarse system equilibrium.

Chaos Theory is the mathematical study of coarse equilibrium behavior.  The fascinating thing about chaos is that it is behavior we are accustomed to seeing, not only in everyday life but, in the universe at large. This is because

There is a universality in the behavior of naturally damped systems when coarsely driven; they all have the same universal mathematical methodology in investigating where else the system can go.

Chaos appears to be the universe’s search algorithm; a mathematically driven way of non-randomly exploring infinite possibilities, and higher dimensionality.

Too long has Chaos focused on the idea of SDIC. Chaos is more than unpredictability. What’s interesting about studying Chaos, is not the unpredictability behavior, but the surprising and highly creative emergent behavior that can result from the coarse synchronicity of internal dynamics.

Chaos Theory is the study of the  Incompressible Mathematics that drives The Surfacing of Incompressible Diversity.