Chaos Theory

Ever wondered why we can predict events in the solar system pretty accurately, why tides are easier to predict, why Halley could accurately anticipate the return of a comet, and yet the weather four days from now is uncertain? This article hopes to provide some insight by exploring the basics of chaos.

Chaos has been considered as the third greatest revolution in 20th century physics(preceded by relativity and quantum mechanics). Initially it was an area of science that was avoided but had always been lurking. For instance, no one could explain the observed unpredictable fluctuations in wildlife populations or the rhythmic quivering of the heart before death. Atmosphere, fluid mechanics, and turbulence all contained unsettled mysteries and some of them are yet to be solved.

It must be noted that one tends to deal with the phenomenon of deterministic unpredictability in most of these cases. This might seem contradictory at first glance since deterministic implies that the future is fully determined by the present and there is no possibility of a different outcome. However, this can be easily illustrated by using a double pendulum when it is moved at large angles.

It still obeys Newtonian physics. However, when two such pendulums are released from the same initial position, the behavior will be entirely different! This is because what we perceive as the initial position isn’t exact for both the pendulums. This is how chaos was encountered by Lorenz, a weather forecaster in the 1960s. The story goes that he had recorded his data up to three decimal places (which then served as the new initial condition) and ran the program. When the same program was run using six decimal places (stored by the computer) the output was different. The discrepancy doubled every four days!

This illustrates that errors in recording the current state of the weather are unavoidable which is what makes weather prediction extremely difficult. Thus, a system is considered chaotic when it shows sensitive dependence on initial conditions. Why is this theory so fascinating? The catchphrase butterfly effect has integrated itself into pop culture: loads of movies, songs, books explore the possibility of insignificant events that change any the future in dramatic and serious ways.

Another reason why we are attracted to it is that unlike relativity or quantum mechanics that seem distant and unreal to most of us (since we don’t deal with objects moving at speeds close to the speed of light or extremely small scales in our everyday life), chaos (though equally bizarre) is seen in everyday things: dripping faucets, jagged coastlines of Norway and Britain, gyrations of stock market, flip of a coin, cotton price fluctuations, even paintings of Jackson Pollock!

However, one gets the notion that if nothing can be neglected, how can we believe in any result? After all, in every engineering course, we start each topic by making assumptions: by neglecting gravity when its effects are small, avoiding friction when possible, by treating objects as point masses and so on. Furthermore, tides are predictable, eclipses can be forecasted. Why can’t we predict weather then? The key is to realize that tides are periodic. Weather isn’t. Hence determinism and periodicity play an important role in forecasting.

Unpredictability occurs after some time called the ‘horizon of predictability’. Weather has a horizon of predictability between four to seven days. For two double pendulums, the value is around one second. For chaotic electric signals, the horizon of predictability is one millisecond.

For the entire solar system, the corresponding number is more than a million years. This means, beyond this time, we don’t know which side of the sun any planet will be, but we know for sure it’ll be in the same orbit. Thus, there is a secret order even in chaos. And scientists visualize this order with the help of strange attractors. An example of the Lorenz butterfly is shown on the bottom of the previous page. Any discussion about chaos isn’t complete without mentioning Henri Poincaré. He had tried to solve the famous three body problem (which contained the essence of chaos).

He used pictures instead of formulas, geometry instead of equations and was probably one of the most gifted mathematicians of the nineteenth century. His phase space viewpoint is so handy that it has been used in areas beyond the realm of science.

Fractals – shapes that once upon a time could be seen everywhere on T-shirts, coffee mugs are sometimes considered the footprints of chaos since they are often linked to chaotic dynamical systems. Fractals are objects that exhibit symmetry under magnification. A perfect fractal can be zoomed in to any scale and it’ll look exactly like the whole. Similar patterns are seen in heart rate variability, internet traffic and earthquake frequencies suggesting that the corresponding laws seem universal, independent of what is undergoing chaos, a sentiment that has been expressed previously by Pythagoras who famously proclaimed that number is the essence of all things.

From an engineering point of view, one of the deepest insights given by chaos and nonlinearity (which gives rise to the subtle order in chaos) is that solutions that look convoluted may be acceptable (and need not be because of any external noise). This elegant field has also given rise to the theory of Lagrangian coherent structures, started by Dr. George Haller in 2000.

This concept has been applied to study the velocity information from the Hong Kong International airport (known for its dramatic and rocky landings because of perturbed air from mountains in the vicinity). Other fields to which this idea has been applied include behavior of pollutants and real-time pollution control algorithms, activities of jellyfish, design of better unmanned submarines, and forecasting course of hurricanes, thereby making it indeed a very strong and powerful means to better understanding of several phenomena in multiple disciplines.

Finally, what are the applications of this discovery? Supporters of chaos theory claim it’ll assist us in getting to the moon with very little fuel. It could have medical applications (eg: predict when epileptic seizures occur). Many claim there is a link between chaos, quantum mechanics, number theory and cancer. They even assert that the mystery of consciousness might be explained by this theory.

Nonlinear dynamics students are probably disappointed because there is no mention of ‘bifurcations’. Chaos enthusiasts are perhaps furious that I haven’t introduced Feigenbaum and his magical numbers. Indeed, there are a lot of concepts, several technical details, significant numbers, and corresponding terms that could be included in this discussion. Instead, I’ve decided to end with a suitable ancient rhyme that expresses the underlying view of chaos theory:

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.

And all for the want of a horseshoe nail.

References
- The Teaching Company: Chaos by Professor Strogatz
- http://www.economist.com/node/14843793
- http://www.whydomath.org/node/space/math_intersect.html
- Chaos. A very short introduction by Leonard Smith