What is Chaos Theory?

20 January 2016:  This page has now been superseded by “Chaos – 101”

Word cloud concept illustration of chaos theory glowing light effect

Chaos Theory is most often associated with the “Butterfly Effect”, but frankly that idea misses the point completely.  Chaos may be unpredictable, but it is also endlessly creative.

Chaos Theory studies creative power of incompressible dynamics…

First of all, forget what you think you might know about Chaos Theory because to-date the conversation has been confused and misdirected, so here we need to take the blinkers off, this will not be like anything you have read before about Chaos…

Part 1 – Linear Dynamics

Coarse Synchronicity

When you trade financial markets you quickly learn that 10% up and 10% down are not the same thing!

If we were to grow $100 in 1000 incremental steps of 0.0095315% we will arrive at $110.  And if we subsequently unwind/decline from this $110 in 1000 incremental steps of 0.0095315% we will arrive back at $100.  If however we were to grow the same $100 to $110 in one incremental step of 10%, and then decline from $110 in one incremental step of 10% we will not arrive back at $100 but at $99 instead.

If we were to grow $100 in 1000 incremental steps of 0.0095315% we will arrive at $110.  And if we subsequently unwind/decline from this $110 in 1000 incremental steps of 0.0095315% we will arrive back at $100.  If however we were to grow the same $100 to $110 in one incremental step of 10%, and then decline from $110 in one incremental step of 10% we will not arrive back at $100 but at $99 instead.

So why is this?  The reason is simple.  Smooth incremental change will synchronize up and down movements, but coarse incremental change will not.


Now you might find this hard to believe but, this simple idea is the cornerstone of all mathematical science. This simple idea is the basis for something called “linear approximation” or “linearization”.  Linearization is used is every area of science; from physics to engineering, from economics to ecology.  By assuming that things changes in a smooth continuous fashion we can approximate the behaviour of any system as being “linear” – which allows us to “compress” this behaviour into neat “solvable” mathematical equations.  Without linearization the mathematics would effectively be unsolvable, and consequently without linearization there would be no mathematical science, which would mean that nothing would be predictable…


The common perception about chaos is that it is all about unpredictability and the so-called “Butterfly Effect”.  The scientific phraseology for the butterfly effect is “Sensitive Dependence on Initial Conditions” (SDIC).  But this phraseology is extremely misleading because it projects the false illusion that everything is fundamentally predictable given the initial conditions.  In reality the use of such phraseology simple reflects the fact that the science of physics is the dominant force in the cult of scientific predictability.

In reality the use of such phraseology simple reflects the fact that the much of science, and physics in particular, is dominated by the cult of scientific predictability.

The science of physics is dominated by linear dynamics.  All the mathematical laws of physics are linear.  But the reason that these “Laws of Physics” are consider to be “Universal Laws” is precisely because they can be linearized, which means we can repeat experiments over and over again, with the same initial conditions, and always get the same predictable result.

In truth however, even this is a little bit disingenuous because nothing is infinitely predictable, but the predictable things are considered to be predictable because it takes a long time for the errors to surface (so far into the future in fact, that we have plenty of time to make the appropriate adjustments).

The reality is that nothing happens in the absence of anything else; but to-date physics has much pretty only focused on those things that happen in the presence on minimal disturbance.  Minimal disturbance means linear dynamics — and linear dynamics are predictable because “linear (approximated) behaviour” is really just another way of saying that a system’s behaviour is “not affected (very much) by feedback”


If we invest $100 in the stock market and achieve a 50% annual return for 2 years, then in the first year we will make $50, and in the second year $75.  The extra $25 in year two in the compounding effect of positive feedback.

50% gains on a yearly basis are nice, but a little unrealistic.  In efficient financial markets such price rises should be ultimately pulled back down to earth because efficient markets are meant to be equilibrium seeking systems, and consequently can be thought of as self-stabilizing, self-correcting, negative feedback systems.

In our previous investment analogy, we saw that by linearizing the growth of our investment, we could reduce the difference in the dollar amount of compound growth (or de-compounding loss) from one step to the next to virtually zero, allowing the synchronization of the up and down movements of our investment.

And so when incremental change driven by positive feedback is smooth and continuous, then negative feedback will be smooth and synchronized.  However when incremental change driven by positive feedback is jumpy and coarse, then negative feedback will be jumpy and de-synchronized.  In other words

coarse positive feedback leads to coarse incremental change which causes desynchronization within negative feedback systems…

Compressible Dynamics

Pretty much all the mathematics in science is based on the fact that in the absence of coarse feedback we can compress behaviour into neat linear equations.  With the exception various forms of friction, physics in general does not deal with feedback.

In the realms of frictionless mechanics, cosmology or quantum mechanics things are easy to linearize because things tend to be under the influence of constant forces.  But even when we do have to deal with friction we are still usually dealing with a constant damping force.  So it is this absence of unknowable future feedback forces that make these systems compressible to neat linear dynamics.

The End of Physics?

For the last 350 years since Galileo rigorous science has rigorously followed “the scientific method”.  The scientific method basically says that a theory is useless unless it is “falsifiable”.  This means that a hypothesis must offer the possibility that experiment could prove it to be false  — or in other words the theory must make predictions that can be tested by experiment.  But as we have just discussed only systems that are not subject to (much) feedback are predictable, which means that every other type of system is unpredictable and consequently not bound by or amenable to the scientific method.

Furthermore, linearization is the reason, despite nature’s obvious progressive evolution, that the laws of physics have no “arrow of time”.  Linearization means that going forwards and backwards in time has no effect as long as we stick to mathematics that employ smooth continuous change.  This doesn’t mean that the laws of physics are wrong, it simply means that the linear laws of physics can, neither explain the behaviour, nor explain the evolution, of complex adaptive systems.

So does this mean we have reached the end of physics?  Well no, it simply means that thus far we have restricted ourselves to investigating and codifying only simple linear systems and simple linear dynamics (and pretty much neglected everything else).   So we have absolutely not reached the end of physics, but we have probably reached the end of the beginning…

Part 2 – Nonlinear Dynamics

Physicists like to describe physics as a “hard science” which of course implies that other sciences are in some way not quite as good, not quite as elevated.  Most science, especially the social sciences, are often referred to as “soft sciences” because their experiments do not produce repeatable results.  In truth however we could say that physics is actually the easy science, and most of the soft sciences are difficult because these so-called soft sciences have to deal with our everyday world where most things are forever buffeted and battered by constantly changing feedback forces and coarse nonlinear dynamics.

Physics in general doesn’t like to deal with nonlinear dynamics because these dynamics are messy and  incompressible, and physics likes to isolated itself from messy and incompressible dynamics.   But in the latter part of the 20th century however, we were to be offered the opportunity to see how these messy dynamics could in fact be our first insight into the creative power of incompressible feedback…

Chaos Theory + Discrete Mathematics

By the early 1960’s the computer revolution had brought with it a computational microscope that allowed us to study coarse incremental change using discrete Recursive Mathematics”.  Before this  time we had been limited to the (linearized) mathematics of “continuous change in continuous time”.  Computers gave us a tool to study the long-term effects of “incremental step-by-step change in discrete incremental time”. 

When Chaos was first discovered there was a lot of excitement, because it seemed to be answering questions that nobody had previously thought to ask.  But gradually over time this excitement waned — most likely because despite the vast quantity of articles and books written on the subject (and its many related cousins) there never materialized, from all this work, a clear and concise understanding of what Chaos Theory is actually all about.  And so over the last 15 to 20 years or so Chaos Theory has found itself pretty much relegated to the minor leagues of science, even to the point of being considered by some as nothing more than an “over-hyped mathematical curiosity”.

Coarse Damping

Unfortunately, the name Chaos Theory and the concept of the butterfly effect has driven the perception that chaos theory is about unpredictable chaos.  Chaos Theory however is not the mathematics of pandemonium and disarray.  Chaos Theory is in fact the mathematical study of positive feedback within self-stabilizing and self-integrating systems.

Chaos Theory shows us that strange things can happen when excessive positive feedback becomes incompressible by the overall negative feedback of the system as a whole.  Coarse positive feedback within negative feedback systems causes coarse-negative-feedback, and this coarse damping” disrupts stabilizing pull of a bland and featureless equilibrium, causing residual behaviour and structure to emerge in its stead…

So, as it turns out the death of Chaos has been greatly exaggerated.  In our evermore interconnected world Chaos Theory is going to be important (still badly named but important nonetheless) because

Chaos Theory explains incompressible feedback and the mathematics of emergence.