The Chaos Theory

The Chaos Theory

When people think of chaos, they think of turbulence or unpredictable outcomes. Yet, chaos is not necessarily a bad thing. According to scientists, chaos is highly sensitive to a set of initial conditions. In this sense, chaos theory can be useful in determining the probability of certain systems responding positively to controlled changes.

  • What is Chaos?: Five lessons on chaos theory.
  • Billiards: An interesting introduction to chaos theory through the science of billiards. 
  • Symmetric Chaos: A picture gallery of symmetric fractals, symmetric icons, chaotic quilts, and others.

Determinism states that all events are caused by a chain of initial occurrences. The universe is believed to be controlled by causal laws so that only one possible state exists at any given time. There are some varieties of determinism including causal determinism, logical determinism, environmental determinism, biological determinism, and theological determinism. Closely tied to Newtonian physics in the West, determinism is associated with an Eastern concept such as Karma.

Does order come from chaos? Henry B. Adams, author of The Education of Henry Adams, famously said, “Chaos was the law of nature; Order was the dream of man.” Neither the law of nature nor the dream of man had been all that kind to Henri Poincaré, the man commonly known as the “Father of Chaos Theory.” Attempting to answer King Oscar II of Norway, he couldn’t prove that the solar system was stable, or otherwise.

Sometime in the 1960s, meteorologist Edward Lorenz was working on a mathematical model to determine the movement of air in the atmosphere. Studying weather patterns, he discovered that changes are quite random. Minute changes in initial conditions would bring about vastly different weather patterns. This came to be known as the Butterfly Effect. Later, his research led to the Lorenz Attractor.

In chaos theory, a small change in the initial conditions of some highly sensitive systems can bring about some complex and random results. It’s rather unpredictable. More apparent in long-term systems, the outcome may not be the same, even if all other factors are held constant. Chaos theory is used in various scientific disciplines encompassing biology, computer science, economics, mathematics, philosophy, psychology, and more.

  • Chaos Theory: A look at the origin of the chaos theory.
  • Chaos UMD: The Chaos Group of the University of Maryland with links to Chaos Gallery, Software, Computing Resource, and others.
  • The Chaos Experience: Good description of chaos theory with links to Chaos History, Real-Life Examples, Chaos Library, and more.

Born in Poland on November 20, 1924, French mathematician Benoit Mandelbrot is called the “Father of Fractal Geometry.” Many people have been interested in Fractal Geometry because of him. In the 1970s, Mandelbrot improved the Julia set of fractals, setting up the basis for fractal geometry. He proposed that fractals are present in clouds, mountains, galaxy clusters, and all of nature.

The Sierpinski’s Triangle is named after Waclaw Sierpi?ski, the Polish mathematician who described it in 1915. It is an excellent example of a fractal, and easily understood. Starting from a single triangle, new triangles are created at the midpoints of other triangles. Since there can be countless iterations, there are infinite number of triangles as well.

The Koch Snowflake is a simple fractal based on the Koch curve. Starting from an equilateral triangle, triangles are added at the trisection points, pointing outward, infinitely. The second iteration is actually the Star of David. By the third iteration, it looks somewhat like a snowflake, and it goes on.

Recursively defined as f(x)=x^(2+c), the Mandelbrot set is known as the simplest nonlinear function. When f(x) is plugged into x continuously, the Mandelbrot set is equal to all generated expressions. For values of c=-1.1. -1.3, or -1.38, the mathematical function is normal, but not for c=-1.9, rendering it chaotic.

With complex fractals, the pictures will be much more complicated. This is done by using geometric shapes instead of lines in the Mandelbrot set, depending on various values. The degree of complexity can be further increased when the system is altered.

As technology advances, the future is increasingly becoming harder to predict. In this climate, the chaos theory is even more relevant. Through chaos, order will ensue.

  • Logistic Map: A very simple form of chaotic process.
  • Chaos and Complexity: Online resource with links to useful sites, interactive sites, discussion areas, and more.