The Secret Life of Chaos
How Does Order Emerge from Disorder?

The Secret Life of Chaos
How Does Order Emerge from Disorder?

It could be argued that the concept of the yin yang in Chinese tradition resembles the union of apparent chaos.

Chaos theory is currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions.

Chaotic behavior can be observed in many natural systems, such as the weather.

It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure 'signal.'

Biologists reason that all living organisms on Earth must share a single last universal ancestor, because it would be virtually impossible that two or more separate lineages could have independently developed the many complex biochemical mechanisms common to all living organisms.

The earliest organisms for which fossil evidence is available are bacteria, cells far too complex to have arisen directly from non-living materials.

The lack of fossil or geochemical evidence for earlier organisms has left plenty of scope for hypotheses, which fall into two main groups:
  1. that life arose spontaneously on Earth
  2. that it was "seeded" from elsewhere in the universe
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy.

Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions; an effect which is popularly referred to as the butterfly effect.

Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general.

This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.

In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.

In common usage, "chaos" means "a state of disorder".

However, in chaos theory, the term is defined more precisely. Although there is no universally accepted mathematical definition of chaos, a commonly used definition says that, for a dynamical system to be classified as chaotic, it must have the following properties:
  1. it must be sensitive to initial conditions
  2. it must be topologically mixing
  3. its periodic orbits must be dense
The requirement for sensitive dependence on initial conditions implies that there is a set of initial conditions of positive measure which do not converge to a cycle of any length.

It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure 'signal.' There will always be some form of corrupting noise, even if it is present as round-off or truncation error.

Thus any real time series, even if mostly deterministic, will contain some randomness. All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system always evolves in the same way from a given starting point.

An object whose irregularity is constant over different scales ("self-similarity") is a fractal. While many things in nature look chaotic, they are not. Scientists are starting to observe that there are patterns to many things around us.

Thus, given a time series to test for determinism, one can:
  1. pick a test state
  2. search the time series for a similar or 'nearby' state
  3. compare their respective time evolutions
Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby state. A deterministic system will have an error that either remains small (stable, regular solution) or increases exponentially with time (chaos).

A stochastic system will have a randomly distributed error. Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test' state.

To define the state of a system one typically relies on phase space embedding methods.

Typically one chooses an embedding dimension, and investigates the propagation of the error between two nearby states. If the error looks random, one increases the dimension. If you can increase the dimension to obtain a deterministic looking error, then you are done.

Though it may sound simple it is not really. One complication is that as the dimension increases the search for a nearby state requires a lot more computation time and a lot of data (the amount of data required increases exponentially with embedding dimension) to find a suitably close candidate.

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions; where a small change at one place in a nonlinear system can result in large differences to a later state.

The effect derives its name from the theoretical example of a hurricane's formation being contingent on whether or not a distant butterfly had flapped its wings several weeks before.

If the embedding dimension (number of measures per state) is chosen too small (less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the dimension too large – the method will work.

When a non-linear deterministic system is attended by external fluctuations, its trajectories present serious and permanent distortions.

Furthermore, the noise is amplified due to the inherent non-linearity and reveals totally new dynamical properties.

Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate the deterministic part risk failure.

Things become worse when the deterministic component is a non-linear feedback system.

In presence of interactions between nonlinear deterministic components and noise, the resulting nonlinear series can display dynamics that traditional tests for nonlinearity are sometimes not able to capture. The question of how to distinguish deterministic chaotic systems from stochastic systems has also been discussed in philosophy.

Benoit Mandelbrot - Hunting the Hidden Dimension

Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child.

Mandelbrot spent much of his life living and working in the United States, and he acquired dual French and American citizenship.

Mandelbrot worked on a wide range of mathematical problems, including mathematical physics and quantitative finance, but is best known as the father of fractal geometry.

He coined the term fractal and described the Mandelbrot set. Mandelbrot extensively popularized his work, writing books and giving lectures aimed at the general public.

Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of Olbers' paradox (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox.

He postulated that if the stars in the universe were fractally distributed (for example, like Cantor dust), it would not be necessary to rely on the Big Bang theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.

Although Mandelbrot coined the term fractal, some of the mathematical objects he presented in The Fractal Geometry of Nature had been described by other mathematicians. Before Mandelbrot, they had been regarded as isolated curiosities with unnatural and non-intuitive properties.

Benoit Mandelbrot emphasized the use of fractals as realistic and useful models of many "rough" phenomena in the real world. Natural fractals include the shapes of mountains, coastlines and river basins.

Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to non-smooth objects in the real world.

He highlighted their common properties, such as self-similarity (linear, non-linear, or statistical), scale invariance, and a (usually) non-integer Hausdorff dimension.

He also emphasized the use of fractals as realistic and useful models of many "rough" phenomena in the real world. Natural fractals include the shapes of mountains, coastlines and river basins; the structures of plants, blood vessels and lungs; the clustering of galaxies; and Brownian motion.

Fractals are found in human pursuits, such as music, painting, architecture, and stock market prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry:

"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."

— Mandelbrot, in his introduction to The Fractal Geometry of Nature

Mandelbrot has been called a visionary and a maverick. His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal Geometry of Nature accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics.

Arthur C Clarke - Fractals - The Colors Of Infinity

Sir Arthur Charles Clarke was a British science fiction author, inventor, and futurist, famous for his short stories and novels, among them 2001: A Space Odyssey, and as a host and commentator in the British television series Mysterious World.

For many years, Robert A. Heinlein, Isaac Asimov, and Arthur C. Clarke were known as the "Big Three" of science fiction.

Early in his career, Clarke had a fascination with the paranormal and stated that it was part of the inspiration for his novel Childhood's End.

Citing the numerous promising paranormal claims that were shown to be fraudulent, Clarke described his earlier openness to the paranormal having turned to being "an almost total skeptic" by the time of his 1992 biography.

During interviews, both in 1993 and 2004–2005, he stated that he did not believe in reincarnation, citing that there was no mechanism to make it possible, though he stated "I'm always paraphrasing J. B. S. Haldane: 'The universe is not only stranger than we imagine, it's stranger than we can imagine.'" He described the idea of reincarnation as fascinating, but favoured a finite existence.

Arthur C. Clarke presents this unusual documentary on the mathematical discovery of the Mandelbrot Set (M-Set) in the visually spectacular world of fractal geometry. This show relates the science of the M-Set to nature in a way that seems to identify the hand of God in the design of the universe itself.

Dr. Mandelbrot in 1980 discovered the infinitely complex geometrical shape called the Mandelbrot Set using a very simple equation with computers and graphics.

Arthur C. Clarke's soft-spoken style sets the "common man" at ease, and his pinpoint commentary makes the concept of fractals easy to understand. One need not be a stellar mathematician to grasp the concepts and why they are profound. The experts are trotted out, and they, too, explain fractal geometry in ways that are accessible to everyman.

Fractals are part of our lives, and maths informs everything that exists, whether natural or man-made.