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

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.

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.]}

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.

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.

Early in the 1980’s there was a lot of enthusiasm about the emerging science of Chaos Theory.  Chaos appeared to be answering questions no one had previously ever thought to ask.

Many thought that Chaos Theory might be “The Next Big Thing”; if Relativity Theory and Quantum Theory were the sciences of the 20^th century, then Chaos Theory and its cousin Complexity Theory were to be the sciences of the 21^st century.

Some thirty years on however, and this enthusiasm has waned somewhat; and while Chaos may have provided some interesting insights into many different and diverse scientific disciplines, it most definitely has failed to live up to, or even come close to, its initial high expectations.

Good science is essentially about simplifying complexity.  It is about showing that each bit of seemingly unrelated complexity is in fact just a unique realization of some more general underlying simplicity.

Good reductive science however has proved somewhat difficult for both Chaos Theory and Complexity Theory.  Much of the reason for this difficulty can be attributed to the fact that despite all that has been written on these subjects (and the many related areas) there has never materialized, from all of this work, a clear and simple unifying theme…

Chaos and Complexity are certainly perceived to be related — different yet in some way similar — but the lack of a unifying theme is likely due to the fact that both subjects have proved a little too hard to pin down and precisely define.  The generally accepted comparison between the two is usually something along the lines of:

• “Chaos is a form of Complex Behavior that can arise in Simple Systems”
• “Complexity is a form of Simple Order that can emerge in Complex Systems”

Theses definitions are acceptable (if somewhat vague) but unfortunately they fail to capture the vital essence of both, and Chaos in particular.

Failing to capture the essence of Chaos, has meant that not only has it not fulfilled its full scientific potential, but on the contrary, it has been over the years virtually relegated to the status of mere mathematical curiosity.

I hope herein to reverse the misfortunes of this underestimated gem of mathematics. My goal is to try and shed some light on the vital essence of both Chaos and Complexity; and in so doing, I hope to reveal that although these fields are different, they are nonetheless very similar; for they are not different as in opposites, but merely different manifestations of the same unifying underlying behavior : a behavior that could be described as “coarse relaxation” or “coarse stabilization”, but probably best described as

“Coarse Damping”.

“Damping” is a term used in engineering to describe the elimination of unwanted vibrations and fluctuations in a system.  Damping usually involves the external dissipation of excessive energy.

Chaos and Complexity however are not about the external dissipation of energy; on the contrary they are  about the internal distribution of energy.

Chaos is about the coarse internal synchronization of competing forces which pull a system spontaneously to various forms of “complex behavioral equilibrium”; and Complexity is about the internal dissymmetry/imbalance of parts, within a system, which pulls the system as a whole spontaneously to a “complex structural equilibrium”.

So Chaos and Complexity are indeed related…

• Chaos is a form of coarse damping in “time”: it is short-term asynchronicity within long-term synchronicity.
• Complexity is a form of coarse damping in “structure”: it is local imbalance with global balance.

Chaos is “Coarse Synchronicity”.

Complexity is “Coarse Entropy”.

• Coarse Synchronicity: A tendency to resist the gravitational pull of a finely-tuned, highly synchronized but fundamentally bland and stable equilibrium.
• Coarse Entropy: A tendency to resist the gravitational pull of a finely-tuned, highly symmetric, but fundamentally bland and featureless equilibrium.
  

So why is any of this important?

It is important because equilibriums are everywhere.  In nature, everything is some sort of system; and all natural systems spontaneously tend towards some form of equilibrium.

Mathematical Chaos shows us that there is however more to equilibrium than meets the eye.  Equilibrium, it turns out, is not always so easy to fine-tune to, and so can come in many “coarse” forms…

Welcome to my Blog.  This blog is intended to act as a repository for my work on

—–   Chaos   &   Complexity   —–

Early in the 1980’s there was a lot of enthusiasm about the emerging science of “Chaos Theory”.  Chaos appeared to be answering questions no one had ever previously thought to ask.

Many thought that Chaos Theory might be “The Next Big Thing”; if Relativity Theory and Quantum Theory were the sciences of the 20^th century, then Chaos Theory and its cousin Complexity Theory were to be the sciences of the 21^st century.

Some thirty years on however, and this enthusiasm has waned somewhat; and while Chaos may have provided some interesting insights into many different and diverse scientific disciplines, it most definitely has failed to live up to, or even come close to, its initial high expectations.

One of my favorite quotes is “If I had more time I would have written a shorter letter”.  The beauty of this famous line is in its self-referential brevity and simplicity.  The point is very clear: brevity and simplicity are very hard to do!…

Writing a shorter letter is, of course, what’s difficult about science.  Good science is essentially about simplifying complexity.  It is about showing that each bit of seemingly unrelated complexity is in fact just a unique realization of some more general underlying simplicity.

You would think that this would apply to Chaos Theory and Complexity Theory.  These subjects are certainly perceived to be related in some way, but even after more than 30 years of study both have proved a little too hard to precisely pin down.  The generally accepted comparison between the two is usually something along the lines of: 

• “Chaos is a form of Complex Behavior that can arise in Simple Systems…”
• “Complexity is a form of Simple Order that can emerge in Complex Systems…”

Theses definitions are acceptable, if somewhat vague, yet they certainly fail to identify, a clear and simple unifying theme…

The goal of this website is to provide the elusive unifying theme of chaos and complexity…

Hopefully – if I can write a short enough letter – the contents of this website will reveal how Chaos and Complexity are merely different manifestations of the same common underlying behavior, behavior that could be best described as “Coarse Damping” to equilibrium.