The butterfly effect

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To properly understand the butterfly effect, we first need to understand Chaos Theory. Chaos Theory is a study in mathematics and it focuses on the study of chaos. There can be seemingly random states of disorder in Dynamical systems that are regulated by laws that are extraordinarily sensitive to beginning conditions or minor changes. Dynamical systems are those whose behavior is defined by a set of rules.

Edward Lorenz, an American mathematician, and meteorologist noticed that minor disturbances, such as the flap of a butterfly’s wings, could affect the events leading up to a tornado. A very small change in the initial conditions had resulted in a significantly different outcome. Lorenz was on his computer one day running global climate models, to save time, he ran one model from the midway rather than the beginning. The two weather forecasts, one based on the full process including initial conditions; and the other based on a subset of the data, beginning with the process already half-completed, were radically different. Lorenz observed that nonperiodic behavior is caused by sensitivity to initial conditions; the more a system varies, the less likely it is to form a recurring sequence.

The butterfly effect applies to systems other than the weather; in fact, any intricate system can be harmed by seemingly minor changes. The movement of asteroids in the solar system, for example, can be hard to anticipate. This is because asteroids’ courses can be influenced by a variety of gravitational influences throughout the solar system, including those exerted by the sun, planets, moons, and even other asteroids.

As Lorenz pointed out, not all nonlinear systems are chaotic, but all chaotic systems are nonlinear. However, chaos is not the same as randomness. The form, known as a Lorenz attractor, demonstrated that practically all chaotic occurrences can only change within certain bounds.

Way back in the 19th century Henri Poincare, a French mathematician, was working on The Three-Body Problem. This problem was worded as such; (a given system of arbitrarily many mass points that attract each according to Newton’s law, under the assumption that no two points ever collide,) try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly. We can translate this simply, as taking three-point masses’ initial positions and velocities and interpreting their subsequent motion using Newton’s Laws of Motion and Newton’s Law of Universal Gravitation.

There was even a reward offered by King Oscar II. Poincare submitted his solution and won the competition. Although shortly after, he realized that he had made an error in his calculations. He never actually ended up proving his solution, but he was the first to lay the foundation what we know today as The Butterfly Effect.

Chaos Theory essentially requires us to toss aside the notion of being able to accurately forecast things, at least for very complex systems. It deals with the nonlinear, which are impossible to predict or regulate with any degree of confidence due to their very nature. We can never be sure about anything in complex systems since even minor deviations in the beginning point can throw the conclusion off dramatically, and flaws in any model, equation, or algorithm add up over time.

Writer: Golda Abs

Editors: Omar Alturki/Uzay Kara

References:

Bradley, Larry. The Butterfly Effect – Chaos & Fractals, http://www.stsci.edu/~lbradley/seminar/butterfly.html. 

“The Butterfly Effect and the Maths of Chaos.” Maths Careers, 21 Feb. 2020, http://www.mathscareers.org.uk/the-butterfly-effect-and-the-maths-of-chaos/. 

“What Is Chaos Theory?” Fractal Foundation, fractalfoundation.org/resources/what-is-chaos-theory/.