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Ralph Abraham

gives it the form that we see. I can’t speak for everyone, but in my experience, this form moves. Now the historic pictures that they show us don’t move. And the mathematicians of fractal geometry have made movies and they don’t move right. So I think that the resemblance between fractals and visuals is very superficial.

I do have a general idea about the mathematics of these patterns. I call them space-time patterns, and they’re fractal perhaps as space-time patterns. But the incredible symmetries, the perfect regularities, I think, are based on some other kind of mathematics. It is called Liegroup actions. And there are reasons why this kind of mathematical structure is associated with the brain. But even if you believed in the internal origin of these patterns in the physical brain and in the Liegroup action approach, some kind of mathematical source could be expected for these visions because they look so mathematical. They have regularity and perfection. How can an image of something perfect appear in the brain? It just doesn’t make sense. So I suspect these visuals are actual perceptions.

RMN: Dynamical systems are arranged by organizing agents called attractors. Could you explain how these abstract entities function and how they can be used in understanding trends in biological, geographical and astronomical systems?

RALPH: Well, attractors are organizing centers in dynamical systems only in terms of long-term behavior. They’re useful as models for processes only when your perspective happens to be that of long-term behavior. Short-term effects are not modeled by attractors but by a dynamical picture called a phase portrait. Its main features are the attractors, the basins and the separatrix which separates basins. Each attractor has a basin, and different basins are separated by the separatrix. It is said that mathematicians study the separatrix and physicists study the attractors, but the overall picture has these complementary things that have to be understood. The separatrix gives more information about short-term behavior, while the attractors determine the long-term behavior. What is most amazing about them is that there aren’t very many. And that’s kind of surprising because there’s so much variety in the world. I would have expected more variety in the mathematical models for the long-run dynamical behavior, but most of them look alike.

RMN: When an attractor disappears due to sudden catastrophic change, the system becomes structureless and experiences a term of “transient chaos” before another attractor is found. How have you applied this idea to cultural transformations?

RALPH: Well, that’s actually a commonly expressed idea which might turn out to be unfounded. People–including me–want to use this aspect of dynamical systems theory called bifurcation theory to model bifurcations in history. History is a dynamical process and it has bifurcations. And here we have a mathematical theory of bifurcations, so let’s try it. That makes sense. But the bifurcations that are known to the theory, as universal models of sudden change in a process, are not usually characterized by this transformation from one equilibrium stage to another through a period of transient chaos. That’s very exceptional in the theory, and I don’t know if natural systems show this characteristic either.

Let’s say you could collect data about a civil war where you had maybe monarchy before and democracy afterwards, and the monarchy was very steady with institutions that you can depend upon, and so was the democracy, and in the middle you were constantly overrun by the troops of one side or the other, or by guerrillas. If this whole history were reduced to data and then you applied the rigorous criteria of dynamical systems theory to these data, and measured the degree to which it’s chaotic, you might find that the monarchy had a chaotic attractor as the model for its data, in the democracy there is also a chaotic attractor of a completely different shape, and in between you don’t have chaos at all; the transient is not transient chaos but is transient something else, or it’s transient chaos but it’s much less chaotic.

You know that heart physiology shows more chaos in the healthy heart and less chaos in the sick heart. I think it’s dangerous to take the casual aspects and implications of these ideas of chaotic theory and start wildly trying to fit them into some preconceived perception of external reality. A better idea is to get some data and try to construct a model. There’s no lack of numerical data about social and historical process. For example, the total weight of mail sent in mail bags from the American Embassy in Russia to Washington, D.C. is known for over a century. Political scientists have an enormous amount of data. I think the serious applications of mathematical modeling to the political and social process will proceed in the numerical realm. The result

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