What is the Butterfly Effect?
“Could a butterfly flapping its wings in Africa create a storm in America?”
When talking about the butterfly effect, everyone remembers the example of an analogy given by the American mathematician Edward Norton Lorenz, meteorologist and at the same time a great contributor to chaos theory (23 May 1917-16 April 2008); A butterfly’s flapping wings in the Amazon Jungle could cause a storm to break out in the USA. Later, many other versions of this example appeared. The butterfly effect, in its simplest definition, is that when small changes are made to the initial data of a system, unpredictable and large consequences can occur.
How Was the “Butterfly Effect” Idea Born?
In an experiment that Lorenz conducted to model the forecast, a very different result was obtained when he entered the baseline data as 0.506 instead of 0.506127. Based on this experiment, Lorenz concluded that a small change in initial conditions could have enormous and long-term consequences. In 1963, in his award-winning essay, Deterministic Nonperiodic Flow, he wrote:
“Depending on the conditions of uniqueness, continuity, and limitation; A central orbit, in a sense, an orbit that does not have temporal properties, is not stable if not periodic. A decentralized orbit; If it is not periodic it is not uniformly constant, and if it is constant, its stability is one of its temporary properties that tends to disappear as time passes. Given the impossibility of precisely measuring the initial conditions and hence the impossibility of distinguishing between a central orbit and a nearby decentralized one, not all non-periodic trajectories are effectively fixed for practical estimation.”
Lorenz revealed that the weather forecast patterns were inaccurate, it was impossible to know the initial conditions, and a small change could change the results too much. To make the concept understandable, Lorenz began using the butterfly analogy and created a graphical model that he called the Lorenz attractor (or attractor). He used three simple equations to construct this attractor and found that even the slightest changes he made to these equations had very different results. This showed how effective the starting conditions were. Later, other towers were also created (such as the Rössler tow and Hénon tow). Lorenz has made one of his greatest contributions to chaos theory with this model. Attractors enable us to understand chaotic systems, that is, they are the mathematical incarnation of chaos. When we look at attractors, we see an order arising from the complexity and we often use it to explain chaos theory.
Lorenz attractor: Lorenz attractor, which can be expressed in a three-dimensional plane, emerged while trying to create a weather forecast model. When he used new values in the equations, he saw that the graph spiraled and never intersected. The system is not stable, does not exhibit periodic behavior, and does not repeat itself. Meanwhile, Lorenz attractor resembles the wings of a butterfly.
Pretty Math Pictures To fully understand the butterfly effect, it is necessary to understand the chaos theory.
We can explain the relationship between them with an analogy; If we think of chaos theory as dominoes arranged side by side, the butterfly effect is the touch of the first stone. Chaos theory is the science of surprises, nonlinear, and unpredictable.
While most of the natural sciences deal with predictable events such as physical and chemical reactions; Chaos theory deals with non-linear events such as turbulence, weather, stock market, which are unpredictable and impossible to control. Chaos theory can be explained by fractal geometry because the underlying logic is the same.
Fractal geometry is the geometry of nature. It allows us to understand nature better. Euclidean geometry was used until the 20th century. Fractal geometry was born when it was not possible to explain nature with linear shapes, triangles, rectangles, and squares. Trees, rivers, clouds, etc. in nature They form fractal shapes and events in nature exhibit chaotic behavior. To understand nature, one must understand fractal geometry and chaos theory. The term fractal was first introduced by the Polish-born mathematician Benoit Mandelbrot (1924-2010) in 1975. Fractals are complex and dazzling shapes that go towards infinity, formed by many geometric shapes that resemble each other from large to small. The Mandelbrot set developed by Mandelbrot is a set that can be transformed into magnificent fractals in a computer environment with functions obtained by using virtual complex numbers.
As a result, the idea of the butterfly effect has become a concept that influences all humanity. People have started to use the butterfly effect analogy not only in scientific phenomena such as the weather, but also in other fields such as economics, psychology, philosophy, and politics. One of the most used and unscientific examples is: In 1905, a young man applies to the Academy of Fine Arts in Vienna and is sadly rejected. This man is Adolf Hitler and when he fails to realize his dreams, he joins the German army. And you know the aftermath.
Of course, many examples can be given scientifically. For example, even a small increase in the amount of carbon dioxide (CO 2) in the atmosphere will have great effects because carbon dioxide gas is a greenhouse gas, which causes an increase in the average surface temperature of the Earth, that is, global warming. The butterfly effect and chaos theory help us understand nature, the world, and the universe. In fact, it helps us to understand the order arising from the disorder of nature and the universe.