352-245-6169 mdthomasdc@gmail.com

Note:  These topics are all connected together for me and it is difficult to separate them out.  This seems to be much the way they work in life too.


The idea of treating complexity as a coherent scientific concept potentially amenable to explicit definition is quite new: indeed this became popular only in the late 1980’s. That what one would usually call complexity can be present in mathematical systems was for example already noted in the 1890’s by Henri Poincare in connection with the three-body problem.

Most often complexity seems to have been thought of as associated with the presence of large numbers of components with different types of behavior, and typically also with the presence of extensive interconnections or interdependencies.

pp.1068-9 Wolfram, Stephen A New Kind of Science Wolfram Media, Inc. 2002.

Non-Linear Dynamical Systems

Isaac Newtons physics provided a way to predict outcomes of many physical events and the power of these equations came to be seen over centuries of time as ˜laws of nature. If all the variables were known, the outcome of an equation could be known in a precise way. A change in one variable caused a proportional change in the outcome. As computers came onto the world scene in the early 1960s, it became possible to study much more complicated systems. Some systems had long stymied modeling; they seemed ˜chaotic. As Gleick noted:

Where chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the turbulent sea, in the fluctuations of wildlife populations, in the oscillations of the heart and the brain. The irregular side of nature, the discontinuous and erratic side “ these have been puzzles to science, or worse, monstrosities.

Gleick, James. Chaos: making a new science. Penguin Books 1987. New York, N.Y. p. 3.

Gleick, in his popular book, Chaos, largely introduced the concept of chaos to the general population. He recounted how Edward Lorenz, a research meteorologist began in 1960, to use computer programs to model the weather. He had developed twelve equations that could create a primitive version of global weather patterns. He could run the program and, as Gleick wrote:

If you knew how to read the printouts, you would see a prevailing westerly wind swing now to the north, now to the south, now back to the north. Digitized cyclones spun slowly around an idealized globe.

Gleick ibid. p. 11.

In 1961, Lorenz decided to re-examine one specific sequence in more detail and re-entered the values into his computer. To his surprise, the results were distinctly different than his first run. He realized that the initial values in the computer memory had been stored to six digits. His read-out, from which he had re-entered the numbers, only gave the first three. This was only a difference of one part in one thousand and Lorenz had assumed the difference would be inconsequential. Instead, he found that it created an entirely different outcome.

This came to be known as The Butterfly Effect. It is known in science as ˜sensitive dependence on initial conditions. In the decades since this initial insight, we have come to understand that measurement is always approximate. It is not possible to precisely account for every variable in complex systems and therefore outcomes are not ever going to be completely predictable. As more complex models were studied, it was found that even though the models would never exactly repeat themselves, there is often an irregular regularity, a pattern, which recurs. Some factor appeared to cause this near-periodic behavior and this factor became known as an ˜attractor. Gleick explained:

The strange attractor lives in phase space, one of the most powerful inventions of modern science. Phase space gives a way of turning numbers into pictures, abstracting every bit of essential information from a system of moving parts, mechanical or fluid, and making a flexible road map to all its possibilities. Physicists already worked with two simpler kinds of attractors: fixed points and limit cycles, representing behavior that reached a steady state or repeated itself continuously.

In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point. That point is the dynamical system-at that instant. At the next instant, though, the system will have changed, ever so slightly, and so the point moves. The history of the system time can be charted by the moving point, tracing its orbit through phase space with the passage of time.

Gleick ibid, p. 134.

The above explanation constitutes a brief non-mathematical description of the topic of non-linear complexity, but was necessary to provide a minimum perspective that can enable us to discuss the particular strange attractor which is the topic of this paper: gravity.