• Arpan Dey

Chaos Theory And Fractals



In this blog, I will be writing on chaos theory and fractals, and their applications. I have already written an article on chaos theory, but nevertheless, I wanted to write a blog on chaos theory on my website. If you have read my article on chaos theory (here), then there is nothing new in this blog for you. If you have not read the article and if you are not familiar with chaos theory and fractal geometry, and would like to know the basic idea, then you should read this blog.


Chaos theory has successfully proven the older ideas about complexity and unpredictability to be incorrect. Indeed, neither do simple systems always behave in a simple way, nor does complex behavior always imply complex causes. Simple systems can give rise to complex behavior, and complex systems can produce simple behavior. Also, most interestingly and importantly, the laws of complexity hold universally, regardless of the constituents of the system.


The edge of a coastline and turbulent flow of the wind on one hand; population growth and weather forecasting on the other. What links and explains so many phenomena is an entirely new science – the science of unpredictability, the science of order within disorder, the science of pattern, the science of chaos. It was difficult to set up chaos as a mainstream science. However, today it is clear that chaos theory is a highly important and practical science. It may seem that everyday objects, like fluids, are well understood. However, on closer inspection, it is clear that even such simple objects show complex, chaotic behavior. Chaos theory is a new science. As James Gleick says in his book Chaos, "Only a new kind of science could begin to cross the great gulf between knowledge of what one thing does – one water molecule, one cell of heart tissue, one neuron – and what millions of them do." Chaos theory, as is evident from the name, deals with complex dynamical systems, which show chaotic behavior, that is, can’t be predicted easily. Chaotic dynamics essentially depend on two things: expansion and recurrence. Most systems that show expansion and recurrence will show chaotic behavior. The rules of complexity, it should be noted are universal, and applies to all dynamical systems, regardless of their constituents.


So far in physics, we have been considering ideal situations and using perturbation techniques to get an approximate result. But the complex Nature around us doesn’t work that way. Prediction is a messy business, and nothing can be predicted with a 100% certainty. But still, prediction is important in modern science. Now, predicting complex behavior using a set of equations is not possible, even in theory. We must run simulations on computers, of such situations, to be able to make a good prediction. And we must use chaos theory. Above all, chaos theory is a new way of looking at Nature. Fractals are intricate and complex patterns, some of which repeat endlessly. You can see patterns everywhere in Nature. Certain flowers always have a Fibonacci number of petals; that is, the number of petals must always be a number from the Fibonacci sequence, starting with 1: 1, 2, 3, 5, 8, 13, 21, 34… (there’s an evolutionary reason for this). Something as beautiful and complex as a fern can also be constructed mathematically. I am talking about Barnsley’s fern.


Alan Turing’s theory about how interesting patterns form basically states that the chances of interesting patterns forming are maximum when there is an interaction between an activator and an inhibitor, diffusion in all directions and a system with a medium size. This is evident when you look at animals. Medium sized animals like leopards and zebras have interesting patterns on them, while elephants don’t. Although there are certain exceptions. And what’s surprising is that many patterns found in biological animals have been reproduced exactly on computers using Turing’s model. Just by using math, and not biology. But yes, complex patterns form on these animals due to biological and evolutionary reasons. (In some fishes, for instance, melanophores move slowly through the body interacting with the neighbor cells, and this gives rise to emergent patterns on the body of the fish.) Evolution doesn’t care about complexity. If an animal can be fitter and more efficient at the cost of becoming more complex, evolution will go ahead with that.


One of the best ways to understand chaos theory is to look at the animal population. Let us assume that the equation x(next)=rx(1–x) represents the growth of a population. Here, x(next) represents the population for the next year, while x is the population for that existing year. r represents a rate of growth, which may change. The term (1–x) keeps the growth within bounds; as x increases, (1–x) falls. If the population falls below a certain level one year, it is liable to increase in the next. But if it rises too high, competition for space and resources will tend to bring it within bounds. Any population will reach equilibrium after many initial fluctuations. The population gradually goes extinct for small values of r. For bigger values of r, the population may converge to a single value. For greater values still, it may fluctuate between two values, and then four, and so on. But everything becomes unpredictable for greater values. The line representing the population-versus-rate function, gradually, though initially single, breaks into two, four… and then goes chaotic.


When r is between 0 and 1, the population ultimately goes extinct. Between r=1 to r=3, the population converges to a single value. At about r=3.2, the graph bifurcates, since at this value of r, the population doesn’t converge to a single value, but fluctuates between two values. For greater values of r, the bifurcation speeds up; and after a quick succession of period doublings soon the graph becomes chaotic. By this is meant that, for those corresponding values of r, the population fluctuates unpredictably between random values, and never exhibits a periodic behavior. However, on closer inspection, it is evident that the graph becomes predictable at certain points, between the chaotic portion. These can be referred to as ‘windows of order amidst the chaos.’ After the initial chaotic region, suddenly the chaos vanishes, leaving in its wake a stable period of three. This, then continues to double – 6, 12, 24 and goes chaotic again...

The chaotic portion of the graph is actually a fractal. A fractal is a complex pattern that repeats endlessly on closing in. On zooming in, it is evident that the chaotic part, in the above graph, repeats the same pattern endlessly. However, a fractal might not always be self-similar, that is, reveal similar patterns on zooming in. Even the coastline of Great Britain is fractal. A fractal, roughly speaking, is a complex pattern that has a measure of roughness.

On further investigations, Mitchell Feigenbaum found that, on dividing the width of each bifurcation section by that of the next one in the above graph, the ratio always converges to a constant value, now known as the Feigenbaum constant, 4.669... What was curious was that, for all bifurcation diagrams, no matter what function has been used, this number remained the same.


One of the most important predictions of chaos theory is that systems with slightly different initial conditions give rise to fundamentally different results. Technically, this is called sensitivity toward initial conditions. The most popular example is the butterfly effect. A butterfly flapping its wings can give rise to a chain of events that might end up creating a thunderstorm in some distant place. This is only an example, and this idea applies to everything in our universe. Tiny changes in the initial conditions produce results that are very different from each other and are, thus, unpredictable.


The applications of chaos theory in weather prediction are widely known. Edward Lorenz wanted to predict weather conditions. He used three differential equations: dx/dt=σ(y-x), dy/dt=ρx-y-xz and dz/dt=xy-βz. In these equations, σ represents the ratio of fluid viscosity to thermal conductivity, ρ represents the difference in temperature between the top and bottom of the system and β is the ratio of the box width to the box height (the entire system is assumed to be taking place in a three dimensional box). In addition, there are three time evolving variables: x (which equals the convective flow); y (which equals the horizontal temperature distribution) and z (which equals the vertical temperature distribution). For a set of values of σ, ρ and β, the computer, on predicting how the variables would change with time, drew out a strange pattern (now referred to as the Lorenz attractor). Basically, the computer plotted how the three variables would change with time, in a three dimensional space. The lines curved out by the computer seem to be ‘attracted’ to two points. Also, in the attractor, no paths cross each other. This is because, if a loop is formed, the path would continue forever in that loop and become periodic and predictable. Thus, each path is an infinite curve in a finite space. (Though this idea seems strange, this can actually be demonstrated by a fractal. Essentially, a fractal continues infinitely; though it can be represented in a finite space.) What is interesting about the Lorenz attractor is that it is, simultaneously, chaotic yet stable. No matter what perturbations the system is exposed to, one always gets back the infinite, complex fractal, which is itself chaotic. A complex system, thus, can give rise to turbulence and coherence at the same time.

We will talk about a very interesting fractal now: the Sierpiński triangle. And how to generate it by playing the chaos game. Three non-collinear points (say, A, B and C) are chosen on a plane, such that they form an equilateral triangle. A random starting point (say, P) is chosen anywhere on the plane. The game proceeds by following certain conditions. A die is rolled. If the outcome is 1 or 2, the point halfway between the points P and A, is marked. Similarly, if the outcome is 3 or 4, the midpoint of the line segment joining the points P and B is marked. For outcomes 5 or 6, the midpoint of the line segment joining the points P and C is marked. As the game continues, the midpoint of the line segment joining the point last obtained, with A, B or C (depending on the outcome), is marked. If this is continued for long enough, the collection of all the points resemble a beautiful fractal called the Sierpiński triangle. The Sierpiński triangle has an infinite length, because the fractal continues infinitely. However, the area tends to zero, since most of it is just empty space and there is no solid surface. The Sierpiński triangle behaves like an attractor. All points on the plane seem to be attracted in a certain pattern, away from the empty triangular regions. Such a system is not sensitive to initial conditions. This is because, no matter wherever we choose the starting point to be, we will always get back the same pattern, provided we plot points as per the rules of the chaos game.

Intuitively, this fractal, with an infinite length, is ‘more’ than a one dimensional pattern, but ‘less’ than a two dimensional figure, since its area tends to zero. In fact, the fractal dimension of the Sierpiński triangle lies between 1 and 2. Yes, fractals can have fractional dimensions. Though this idea seems absurd, the dimension of a fractal is basically a measure of its roughness. (It should be noted that a fractal doesn’t always have fractional dimensions.) If a one dimensional line is broken into two equal halves, that is, if it is scaled by one half; its mass is also scaled down by one half, since two such halves will reproduce the original line. Similarly, if we scale the side of a square by one half, its mass is scaled by one fourth, since it takes four squares (each of a length one half the length of the original square) to reconstruct the original square. One fourth is just one half raised to the power of two, and this number is the dimension of the square, which is two. Similarly, just as a line is one dimensional and a square two dimensional, a cube is three dimensional because if a side of the cube is scaled by one half, the mass is scaled down by one eighth (or one half raised to the third power), and it takes eight copies of the smaller cube to generate the original cube. For a Sierpiński triangle, on scaling it by one half, we get a similar, but smaller pattern, three of which, when arranged in the right pattern, give back the original triangle. Thus, the mass has been scaled by one-third. Following the above line of reasoning, this means that one half raised to the power of (say) x, should equal one third. This x is the dimension of the Sierpiński triangle. This gives x≈1.585, which is the fractal dimension of a Sierpiński triangle. The ‘chaos game’ may be played with more than three points, to generate more complex fractals. Another interesting fractal is the Sierpiński carpet. It can be generated by dividing a square into a 3X3 matrix, that is, nine squares and then removing the middle square. This operation is then repeated on the eight remaining squares, and so on infinitely. When this same activity is carried out on a three dimensional cube, the Menger sponge is formed. Interestingly, it has an infinite surface area but zero volume.


Now, we will look at some applications of chaos theory. Mathematics, in particular the study of nonlinear dynamics, has successfully addressed questions that biology had failed to do. So chaos theory has many applications in the medical sciences.


Before going to the medical applications, I would like to discuss a very interesting possibility. You might know that holograms play an important role in modern physics. Holography can explain and link two entirely different kinds of theories in physics. It can take us a long way toward a theory of everything, and is one of the most notable ideas in fundamental physics. Holographic duality is also referred to as AdS/CFT correspondence, where AdS refers to Anti de-Sitter spaces, while CFT refers to Conformal Field Theory. All this is technical, but the concept is really simple. Imagine a sphere. CFT is related to the boundary, while the AdS space sits inside the sphere. Everything in the AdS space has a counterpart in the CFT boundary. It is crucial to understand that a hologram is two dimensional, but it can contain all the information about all three dimensions of the object it represents. Think of it like this. A three dimensional universe contains black holes and strings governed solely by gravity, whereas the two dimensional boundary of this three dimensional universe contains ordinary particles governed solely by standard quantum field theory. The interesting possibility is that holograms can have fractal properties. And we have already seen that the universe may be holographic in nature. If the boundary, and therefore the universe’s quantum mechanical interactions are fractal in nature, this can explain quantum entanglement. Particles become entangled as they are the self-similar repeating patterns of one another.


Quantum pioneer Erwin Schrödinger, so many years back, had already formed the then-unusual idea that life was both orderly and complex. He saw aperiodicity as the source of life’s special qualities. In his book Chaos, Gleick writes, "Pattern born amid formlessness. That is biology’s basic beauty and its basic mystery. Life sucks order from a sea of disorder. Schrödinger, the quantum pioneer and one of several physicists who made a non-specialist’s foray into biological speculation, put it this way forty years ago: A living organism has the “astonishing gift of concentrating a ‘stream of order’ on itself and thus escaping the decay into atomic chaos.” To Schrödinger, as a physicist, it was plain that the structure of living matter differed from the kind of matter his colleagues studied. The building block of life – it was not yet called DNA – was an aperiodic crystal. “In physics, we have dealt hitherto only with periodic crystals. To a humble physicist’s mind, these are very interesting and complicated objects; they constitute one of the most fascinating and complex material structures by which inanimate Nature puzzles his wits. Yet, compared with the aperiodic crystal, they are rather plain and dull.” The difference was like the difference between wallpaper and tapestry, between the regular repetition of a pattern and the rich, coherent variation of an artist’s creation. Physicists had learned only to understand wallpaper. It was no wonder they had managed to contribute so little to biology. Schrödinger’s view was unusual. That life was both orderly and complex was a truism; to see aperiodicity as the source of its special qualities verged on mystical. In Schrödinger’s day, neither mathematics nor physics provided any genuine support for the idea. There were no tools for analyzing irregularity as a building block of life. Now those tools exist."


Other than helping us understand life, chaos theory has numerous applications in medicine. The idea is that mathematical tools could help biologists and physiologists understand the complex systems of the human body, without a thorough knowledge of local detail. Chaos theory successfully explained the sudden, aperiodic and chaotic behavior of the heart, called ventricular fibrillation. According to chaos theory, the fibrillation is the result of a disorder of a complex system, like the human heart. Though all individual parts of the heart seem to work perfectly, yet the whole system becomes chaotic, and fatal for human life. (This intuitively shows that the reductionist approach doesn’t always work in science. Often, it is the entire system as a whole that is to be considered, instead of breaking it down to smaller and smaller parts. I have discussed reductionism and emergence in one of my previous blogs.) Ventricular fibrillation is not a behavior that returns to stable conditions on its own; rather this fibrillating state is itself stable chaos. Fractal geometry also allows the formation of bounded curves of great lengths, and that is how the lungs manage to accommodate so large a surface area inside so small a volume, which in turn, increases the efficiency of the respiratory system. Fractal geometry has also been used to model the dynamics of the HIV virus, which is responsible for AIDS. Bone fractures are fractal and even the surface structures of cancer cells display fractal properties, and this property can be manipulated to detect cancerous cells at an early stage. Fractal patterns exist throughout the body – from the tissues to the way blood vessels branch. So these are some of the applications of chaos theory in medicine. Did you know that chaos may be intimately connected to psychology as well?


Nigel Lesmoir-Gordon writes in his book Introducing Fractal Geometry, "It is entirely conceivable that the low level of fractal complexity in modern inner cities is a strong contributing factor to the high incidence of depression reported in these kinds of environment." This may be why we still are fascinated by the complex architecture of ancient times.


If the link between chaos theory and biology is not clear to you yet, let’s consider an interesting question. How does a plant grow in one direction? What’s so surprising about that? Because a plant has to prevent its parts from growing in all directions (which is natural) and grow in a particular direction. It has to prevent its cells from bulging and direct the growth in one particular direction. How does it achieve this feat? In plants, the microtubules grow in a particular direction. When microtubules collide perpendicularly, one microtubule simply shrinks. If they collide at a slanted angle, they simply rearrange themselves in one direction. If you run a computer simulation with lots of microtubules, over time you see that they arrange themselves in one direction. So, interaction between the microtubules gave rise to the desired effect. It gave rise to organization in the system. And for consciousness to emerge, some degree of organization is required. We need complexity, we need chaos, but we need coherence, order in the chaos as well.


One of the most perplexing problems that remain is the fact that consciousness is stable. It is evident that consciousness is an emergent phenomenon that arises out of the combination of and interactions between many units in a big and complex system. When many units interact, there is a probability that some special effects and combinations will arise, with a greater-than-the-sum effect. But, why does this special state remain in the same state for so long? Why does it not collapse? The best answer to this might lie in the study of nonlinear dynamics. Chaos theory explains such things in terms of mode locking or entrainment. In Chaos, Gleick writes, "This phenomenon, in which one regular cycle locks into another, is now called entrainment, or mode locking. Mode locking explains why the Moon always faces the Earth, or more generally why satellites tend to spin in some whole number ratio of their orbital period: 1 to 1, or 2 to 1, or 3 to 2. When the ratio is close to a whole number, nonlinearity in the tidal attraction of the satellite tends to lock it in. Mode locking occurs throughout electronics, making it possible, for example, for a radio receiver to lock in on signals even when there are small fluctuations in their frequency. Mode locking accounts for the ability of groups of oscillators, including biological oscillators, like heart cells and nerve cells, to work in synchronization. A spectacular example in Nature is a Southeast Asian species of firefly that congregates in trees during mating periods, thousands at one time, blinking in a fantastic spectral harmony." Simply put, when two chaotic systems couple, it results in synchronization and stability. Another example is that when the audience starts clapping, although initially everyone claps at their own pace, after a few seconds, the clapping spontaneously becomes synchronized and everyone (unconsciously, of course) claps together. Imagine a chaotic system, consisting of a huge number of simple units, which interact with one another. The system, as a whole, is chaotic and unpredictable. However, in subsystems within the system, predictable and ordered behavior may arise (due to the formation of loops, say, etc.). This order in the system makes it stable, for a long time. However, ultimately, chaos takes over. If there was no regional order, and the system was chaotic throughout, then there would only be bursts of consciousness, and no stable life that can evolve further.

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