044 045 Chaos Theory: The Science Behind the Butterfly Effect Pratham Bhargava It’s been in movies, blockbuster films and even became a meme at one point, but the message has been relatively simple: small changes in life can often lead to greater consequences. It only took a small change for the MIT professor Edward Lorenz to even come up with the butterfly effect, a term he coined in 1960 whilst working on a computer atmospheric model to help meteorologists predict the weather. Predicting things was relatively straightforward in the 1700s. Science, and in particular mechanics, made predicting how an object would move easy. Newton’s laws of motion and universal gravitation meant you could calculate how fast something would drop to the ground, or even how fast the Earth was orbiting the Sun. It all came down to the idea of determinism, that the future was already there, we just had to wait for it to happen. Take a car for example. If we know it is accelerating at 1ms-2, then it makes sense that in 10s, the car would be travelling at 10ms-2. However, there was one problem. Or more accurately, a problem involving three bodies (known as the three-body problem). The motion of two interacting bodies was easy enough but add one more and Newton himself was unable to find a solution to how three bodies interacted after some time with the initial sets of velocities and positions. Later on in the 1800s, the French scholar Pierre Laplace speculated rather philosophically on predicting the state of the universe. Using Newton’s laws of motion and universal gravitation, he speculated that if a person were to know the exact position, velocities and momenta of all the particles in the universe at any given time then this person would in fact be able to calculate the positions, velocities and momenta at any time. Therefore, ‘Laplace’s Demon’ as commonly referred to by scientists, would be able to know the state of the universe at any time. Of course, if the car we used earlier was accelerating at the same rate for 10s then you would be able to calculate how far it travelled in 8s, 9s, 11s, or 20s. The problem is there probably isn’t someone who could take a screenshot of the universe. And if we can’t do that, is the universe even predictable in the first place? In the 1960s while Edward Lorenz was experimenting with his computer model of the weather, he accidentally (like many discoveries in science) noticed something strange with the results the computer had churned out. The program was a simple model of convection currents in the atmosphere, and for each variable influencing how warm air expands, he made an equation. Lorenz would input a value, and the computer would compute a set of values which essentially describe how the weather would look in numbers. However, after taking a break, he chose to input the figures the computer listed halfway into the program (which were to three significant figures) and surprisingly got completely different weather descriptions. The exact results which he had input earlier had spiraled into a string of entirely different numbers. As anyone would do, he checked if his computer was broken. It wasn’t. He tried again. Same result. In fact, Lorenz after some persistence simplified his equations into three differential equations. But unfortunately (or fortunately for chaos theory), he got the same result. What Lorenz had discovered was proof that the weather was a chaotic system. It wasn’t completely random but deterministic – only if you could input the exact same conditions would you get the same result. And the results Lorenz had initially input into the computer were to 6.s.f. while the later ones were rounded to 3.s.f (so the new inputs were very slightly different). This is a key principle of chaos called sensitive dependence to initial conditions. In the same way, our solar system is a chaotic system too. If we go even 10 million years into the future, planets are predicted to completely change orbit just because of this spiraling effect. Newton couldn’t figure out the three-body problem because it was an example of chaos – some systems are impossible for us to predict to an infinite precision because we can never know their initial conditions exactly. How could weather forecasts even be slightly reliable then? The trick is, as we will see scientifically later, that over longer periods of time chaotic systems can be easier to predict. That’s why it’s much easier to predict the climate (the average weather over a long time) instead of the weather. Ensemble forecasts are made for the weather instead (the initial conditions are changed slightly to produce a range of forecasts for meteorologists to analyze). Of course, general predictions based on some knowledge of pressures and atmospheric circulation can help, but even super computers won’t be able to give exact weather, especially over more than one week. Now that we understand what a chaotic system is, we can try picturing it. For this, we need to use a 2-D space called phase space. Imagine a pendulum. This is something we can easily predict the motion of using classical mechanics, but it will get interesting with two. If we plot the angle of the pendulum on the x-axis and the velocity on the y-axis and let it go, it will eventually slow down and stop. In our phase space, this will look like an ellipse spiraling towards the origin (since the velocity is 0 ms-1) where the origin is something called an attractor. If there was no resistance, then a closed loop would form (since the velocity would never decrease over time). Closed loops tell us for a specific set of conditions, there is only one future. It’s not like the pendulum will suddenly change velocity. If we try plotting the movements of two pendulums joined together then it’s actually impossible for the pendulums to follow the same path every time. This is again an example of a chaotic system, even though we can easily understand one. Now perhaps the most coincidentally interesting part of phase space is that if we plot Edward Lorenz

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