If you have seen the movie ‘The Butterfly Effect’ or even heard about this phenomenon, it means how tiny changes in big systems can have complex results. I, hereby, present to you the famous quote in this context.
Something as small as the flutter of a butterfly’s wing can ultimately cause a typhoon halfway around the world.
Now, systems in this context can be anything from weather patterns to how big groups of asteroids move to our irregular heartbeats. From a physics point of view, we can look at the ‘Double Rod Pendulum’.
The credit for discovering the ‘Butterfly Effect’ goes to MIT mathematician and meteorologist ‘Edward Lorenz’. Lorenz was solving a set of nonlinear ordinary differential equations on an old computer. At first, he was using initial conditions correct up to 6th decimal places. But due to his computations taking too much time, he decided to use initial conditions correct up to 3rd decimal places and he ended up getting results that really got him surprised. It is quite logical to think that if you change a little at the beginning, it is going to change a little at the end. But these systems don’t behave that way. You can see from this figure how different his results turned out to be and he wrote his seminal paper ‘Deterministic Non-Periodic Flow’ on this research. And it was a very interesting event that led to the coining of the term ‘Butterfly Effect’. In 1972 at the 139th meeting of the American Association for the Advancement of Science, Philip Merilees concocted “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” as a title. And that was how the term ‘Butterfly Effect’ came into existence. And Lorenz’s famous results, when plotted in its phase space projection, gave exactly a butterfly shaped pattern and hence the term.
But the question remains “How do we study Chaos Theory?”. Edward Lorenz himself said one thing about Chaos and that is — “Chaos: When the present determines the future, but the approximate present does not approximately determine the future.” So then, how do we study it? Take Nature for example. The left hand side figure is that of a Romanesco Broccoli and the right hand side is of the portion of the map of a river basin. These pictures, you see, may seem random but it is governed by rules and that’s why Nature makes shapes like these. Rules of Biology and Geology created these patterns. These are known as Fractals.
And it is our understanding of the mathematics behind these rules that will give us an advantage to understand how these complex systems evolve in time. Talking of Physics, Fractals like ‘Hofstadter’s Butterfly’ and ‘Sierpinski Gasket’ (picture attached above) has already been observed experimentally. The former explains the ‘Dynamics of Electrons in a Magnetic Field’ and the latter has been seen in ‘Optical Mesh Lattice’. And the practical applications of Chaos Theory are huge. From understanding the electro-encephalogram of our brainwaves to our interactions with people on a daily basis, how gas moves around in the atmosphere, geological processes of our own Earth, etc. In fact, a study in the ‘Journal of Family Psychology’ attempted to predict divorce rates using Chaos Theory Math from a sample size of 95 couples and they were correct up to 87% of the time. On the other hand, climatologists keep getting better and better with time at predicting weather patterns since the huge data they accumulate over time gives them a better edge at predicting our chaotic weather patterns.
And it is, in fact, our understanding of ‘Order in Chaos’ that will give us the power to predict events arising in the most complex of all systems in the long run.