The science we study at high school some times gives us the impression that (almost) every phenomenon in the world is perfectly understood by the wall of knowledge that humans have built. However, there are times when we have the feeling that something is not quite right. I too one day encountered this feeling as i walked out of a physics class. There was this tinge of acquiescence that I had begun to experience whenever I learned high-school physics; it felt as if the truth of reality was being settled too easily.

As I looked around for means to settle this new-born discomfort, I came across a “theory” which affirmed skepticism. This “theory” popularly called the Chaos Theory, had at its core the idea that the world is not as easily determined as suggested by Newtonian science. The Newtonian revolution in the scientific world had brought about the perception that the world is a gigantic machine demonstrating clockwork. It asserted that nature follows a particular set of laws and we had to simply discover these to decode the magnificence and strangeness of nature in its totality. However this view, which some may call too simplistic, has already been twice run over by revolutionary scientific paradigms: the theory of Relativity and Quantum theory. And now more recently, the chaos theory has dealt another blow to the deterministic nature of the Newtonian paradigm of science. But chaos does hint that the way the world functions has a  different sort of structure : different kind of clockwork lies hidden within. To detect these subtle patterns in the functioning of the world, which apparently seems random, chaos provides us with the required tools. Though this may seem to be a contradictory statement to make, the significance of it will become clearer later in the article.

Chaos differs from the nature of Relativity and Quantum theory in a significant way. Relativity shows its effects at high energies and velocities and quantum theory concerns itself with phenomena which are submicroscopic. On the other hand, Chaos has a much wider domain of application, indeed it can be seen in the most simple of experiments.

In the nineteenth century, these was a thick insulation which existed between mathematicians and physicists. However, around the world there were pockets of experts in the fields of biology, meteorology and especially mathematics and physics whose ideas and thinking were directly associated to Chaos. Gradually this wall between the physicists and mathematicians was dissolved as one understood the need of the other in a obtaining a clear picture of the world. And when this interface did finally take place, Chaos was born. The fascinating history of the development of Chaos has far too many characters to be described here. We will instead look at some of the core ideas of Chaos.

Chaos studies chaotic behavior , i.e., behavior occurring in a class of systems called dynamical systems.

Systems which keep changing…

Simply put, systems which keep changing with time are given the apt name of dynamical systems. A dynamical system consists of a set of possible states, together with a rule which determines the present state of the system in terms of past states. Take the example of a bacteria population in a lab: suppose the rule which determines the population of the fish is f(x)= 2x, where x is the number of bacteria in the culture. Also suppose that at t = 0, the population is 100. After one hour, the population will be f(100) =2*100 = 200, after two hours it will be f(200) = 2*200 = 400 , and so on. Here the bacteria population is determined after each hour by the rule f(x). Other examples of dynamical systems include a swinging pendulum,water flowing in a pipe and weather. Below shown is demonstration of chaotic behavior in the double pendulum system.

So what’s the problem with these dynamical systems ? Well, for one thing the rules governing real-world dynamical systems are not as easy as the above described f(x). Why is this? It is because of the huge amount of non-linearities which exist in reality.

To understand this better, take the case of a population biologist studying a population of fish in a pond: she will try to model the fish population using a rule. However, in her task of finding this rule that governs the fish, she will right away come up against great complexities: to accurately model the fish population she will have to take into account competition from other organisms(e.g. frogs, storks) climate, available food for the fish, diseases and … . This list can get bigger and bigger. Now the problem with a dynamical system is clear: the system has too many external factors affecting it which makes it difficult to accurately model it. To study such unwieldy systems the tools offered by Chaos are essential.

The Butterfly Effect

Feynman had once said,

Physicists like to think that all you have to do is say, these are the conditions, now what happens next?

So what is the problem with such thinking? Well the butterfly certainly is!

The butterfly effect is one of the central ideas associated with the theory of chaos. It was coined by the meteorologist Edward Lorenz in a lecture titled “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”. As the name suggests, it has something to do with a butterfly; the butterfly effect is the concept that a butterfly flapping its wings in New Mexico can cause a hurricane in China. This is simply a metaphor for the idea that even small changes form a significant part of the weather condition, and it is impossible to accurately predict the weather without taking into account these factors that can affect a given weather system.

Technically also known as “sensitive dependence on initial conditions”, this idea more generally asserts that a small change in the initial conditions of a dynamical system can cause great changes in the end state of the system. Take the example of the double pendulum above; releasing the pendulum from a slightly different point and a slightly different velocity will result in a completely different motion. Another example is that of an avalanche: a small input like a loud noise, or a particularly strong burst of wind can cause an avalanche in which huge amounts of energy are transformed. The idea of the Butterfly Effect as mentions even in folklore:

“For want of a nail, the shoe was lost;

For want of a shoe, the horse was lost;

For want of a horse, the rider was lost;

For want of a rider, the battle was lost;

For want of a battle, the kingdom was lost !”

Though this idea of the “Butterfly Effect” may seem like a far away concept, it can be illustrated by a very simple example of the function 2x^{2} - 1 too.If I take the initial value of 0.54322 and iterate it, I get the list in which the first and last few values are shown below.


Even though the first few values quite similar, the end result with the first case is positive while the end result in the second case is negative! The small difference in initial values magnifies after being repeatedly iterated. This is the Butterfly Effect in action.

Graphical Representation of a dynamical system

As it is becoming evident, dynamical systems seem to be monsters. How can these erratic systems be tamed, or to begin with: even understood? To gain insights into the behavior of a dynamical system, scientists sometimes use the graphical representation of the system.These representations are called phase diagrams. This representation is obtained by plotting the state of the system at each instant in time.

Take the example of a simple pendulum: a phase diagram of a pendulum is shown in an ideal situation, i.e., forces such as air drag are not taken into account.


Phase diagrams of an ideal and non-ideal pendulum.

To determine completely the state of the pendulum at a particular instant all you need to know is its velocity and position. By plotting the velocity of the pendulum along the y-axis and position of the pendulum along the x-axis, you get a phase diagram of the system.This would match our observations of the pendulum too: at the extreme position the velocity of the pendulum is zero while at the mean position the velocity is maximum which is shown by the maximum amplitude of the phase diagram at x=0. On the right a phase diagram of a pendulum is given after taking into account frictional forces. Here it is again easily explicable: since the velocity of the pendulum keeps decreasing due to frictional forces, it comes to rest after some time, shown by the spiraling of the graph towards v=0 & x=0. Now if you left the pendulum from a different position or with a different velocity, it will even then eventually come to state of rest. Therefore the point of rest “attracts” the state of the system, and so it is called an attractor.


Attractors are states of a system towards which the system tends to go regardless of the initial conditions. The example of the non-ideal pendulum illustrates the fact that the pendulum tends to reach a particular state regardless from where you release it, i.e., it comes to rest after some time. Such an attractor is called a point attractor, since the system always reaches the same end point. Human life too is a point attractor! We all have the same end state i.e., we die.

Another type of attractors is the periodic attractor. The phase diagram of the ideal pendulum is an example of a periodic attractor since the system settles into a cycle and it has a certain period.

The next important type of attractors is the chaotic type also known as a strange attractor .. This is the kind of attracting point that occurs in chaotic dynamical systems and is of current interest to us. It is called “strange” because it consists of components which cannot be constructed through simple geometric figures such as lines, circles, squares, etc.

The Lorenz Attractor is an example of a strange attractor. It resembles a butterfly’s wings. This attractor came to be when Lorenz was modeling the earth’s atmosphere. In his simple model, he had three variables of pressure, temperature and humidity governed by three equations. When he ran his model and then plotted the results in form of a phase diagram, what he got was this strange attractor. Shown below are the evolution of Lorenz’s weather model at different stages. If you start with 8 random initial values, that is if you choose values for the 3 variables( pressure, temperature and humidity) for 8 different points and let the equations run on these 8 different values, what you get is shown below.

This illustrates two important ideas: even though the starting conditions were quite close each other, the path they follow is completely different. Why is this? The “Butterfly Effect” is in operation. The second idea that it shows is that even though the weather model is a complex system, all the different possible weather results lie on the plane of this figure. Though there is no apparent pattern in the behavior of the weather model, it has this hidden structure behind it. Therefore, despite our inability to predict the definite behavior of the system at a particular instant, we can definitely say that it will lie somewhere on this figure. And so from such attractors, we get some clue about the behavior of these complex dynamical systems. This was what I meant when I said at the start that even more fine order lies within apparent randomness.

Take the simple example of the function 0.8x + 1 . If you iterate the function for 0.5 and 4 , after about 100 iterations you end up with a value that is same to about 9 decimal places!I tried it with 1000 and I astonishingly again got a very close value of 5 ! If you try it with any other values too, the result is surprisingly close to 5.



The discovery of attractors led to further inspection of this strange sort of order, until another more surprising revelation was made: if the phase diagram of the Lorenz attractor is allowed to evolve indefinitely, it would have an infinite number of lines. This is because of the very chaotic nature of the system; no same set of states is ever repeated and so the any one path in the phase diagram is never again traced out. The strangeness of this geometry of the strange attractor baffled scientists for quite some time until slowly the concept of fractals emerged.


Fractals are geometric shapes displaying self-similarity. This means that zooming into a fractal doesn’t change the way it initially looked! Below shown are some very famous examples of fractals.

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These kinds of geometrical figures were first conceived by Benoit Mandelbrot, an American mathematician. The idea of fractals were discovered by Mandelbrot when he noticed a striking common feature in various situations around him.

Engineers at IBM had been for long grappling with inexplicable errors in the transmission of their signals. Mandelbrot had been hired by IBM to explain and solve the problem of periods of errors in the signals which were being transmitted. When he closely analyzed the signals, he noticed something very astonishing. Within the periods of errors in the signals, when suitably magnified, there were periods of undisturbed signal , and within these undisturbed periods there were periods of disturbance and so on. In fact, Mandelbrot found a consistent geometric relationship between the bursts of error and error-free transmission. This immediately reminds us of the concept of fractals. Thus with the acknowledgement that the signals showed fractal-like nature, instead of trying to completely eradicate the disturbances, the engineers could use an alternative strategy to reduce these bursts of error.

Another place where Mandelbrot encountered fractals was in the cotton prices of 1963. Mandelbrot found when he studied the cotton prices that they did not follow the normal distribution i.e., the “bell curve” instead the prices had were extreme variations. This was contradictory to the prevailing ideas of economics. When he analyzed it, he slowly realized that the cotton prices exhibited a different kind of order: they were indifferent to the time scale; the curve for the price change over a day was similar to the curve of the price change over the month! This property of self-similarity naturally suggests the idea that the cotton prices exhibited fractal-like nature and thus Mandelbrot had found a different sort of model for the price changes.

These are some of the very key ideas related to Chaos. There are a multiplicity of real world situations where Chaos helps explain seemingly inexplicable phenomena.Some of the questions that Chaos helps answers are:

How do you predict weather weeks or months beforehand when it shows no pattern? How can you predict irregular stock markets and global economies? Why are human heart beats irregular? How do you predict the population of a species which shows a random pattern? How do fluids flow and how can their motion be predicted? What is the nature of turbulence? How do you program robots to perform random movements? How do you predict the rate of a chemical reaction? How do you predict the election polls? How do you interpret the relationship between employees and the job market? …

This falls in very much with what James Gleick says in his book “Chaos”(which is a highly recommended read),

Now that Science is looking, Chaos seems to be everywhere.

In the coming posts, as I myself learn more about Chaos, I will further illustrate the application of Chaos in specific fields of study and explore the related Mathematics in more detail. Suffice it to say : without Chaos, chaos would reign supreme!


One thought on “Chaos

  1. Pingback: Nonlinear Time Series Analysis | Math In Operation

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