The Butterfly Effect – Chaos Theory

Does the Flap of a butterfly’s wings in Brazil Set Off a Tornado in Texas?

In 1961, an MIT Meteorologist named Edward Norton Lorenz was working on weather simulation equations which were based on 12 variables like temperature, pressure, wind speed. He was trying to predict the weather weeks ahead by using Punch Card Computers. Variation in those 12 variables will provide the state of the weather. For technical reasons, he had to re-run the simulation by providing the initial values of the variables and took a coffee break. When he returned, the output values were vastly different from the previous calculation.

Lorenz thought the computer was malfunctioning but after a brief check, he realized that the entered values were rounded off by just 3 decimal values while the computer calculated for 6. He was shocked that just a fraction of a decimal is predicting totally different weather. Later he reduced the equations to just 3 with 3 variables and made a run. The results were Chaotic. A small variation in one of the variables is resulting in a different state of weather. The problem with Chaotic systems is that we can never measure something with infinite accuracy.

Sensitive Dependence on Initial Conditions

The name Butterfly Effect comes from the title of a paper published in 1972 by Edward Lorenz. Butterfly Effect states that “Even an insignificant change in the initial values of a Chaotic system results in a vastly different output”. It got the attention of the crowd because it narrows down the idea of predictability. We spend decades and billions of money on science to just predict the future but Chaos Theory puts limitations to that. Ever wondered why we can predict Eclipses far away into the future but struggle to predict the weather just weeks away? Before Chaos Theory everything other than the atomic scale was predictable. They had all kinds of formulas and theories to predict the motion and behavior of objects with Newtonian Physics.

Everything from throwing a ball to the motion of planets, everything is predictable. There are some exceptions like the three-body problem under the influence of gravity. Calculating the motion of two bodies is simple but the same motion with three bodies is complicated. Another example is the double pendulum, unlike a traditional pendulum, a double pendulum has another pendulum attached at its end. We can see Chaos in action by releasing two double pendulums simultaneously.

Even though they are released at the same time with the same height, their paths diverge. This is due to the fact that both pendulums have a different coefficient of friction values and have slight variations in their masses and so on. The thing that we don’t realize is that Chaotic systems are everywhere but they are not chaotic on a human time scale. Even the motion of planets is not predictable very far into the future.

Chaotic Systems are not purely random. If we could calculate the exact values, we can predict both the past and future of a system. Lorenz and his team went on and calculated the states of the system and plotted a 3D graph.

The Equations Lorenz used were 3 differential equations with respect to time. The Equations are
• dx/dt = σy – σx
• dy/dt = ρx – xz – y
• dz/dt = xy – βz
Where σ, ρ and β are system parameters and x, y and z are the positions in 3D coordinate system.
σ = 10, β = 8/3, ρ = 28 ; x = 0.01, y = 0, z =0.

After plotting the graph, they realized that the system is attracted towards two invisible centers and they never revisit the same point. This shows the deterministic behavior for a Chaotic system. Chaos Theory is not just about randomness but the predictable behavior of the whole system. Strangely it takes a shape of a butterfly with an infinite curve in finite space. They named this shape The Lorenz Attractor.

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