Stroom 21, (Faculty of Natural Sciences and Astronomy, Univ. of Amsterdam, March 1992)

Interview with Predrag Cvitanović

# Nonlinear dynamics: New tools are found, little ideas work"

### Hanna Brands and Frederique Harmsze

Last November Predrag Cvitanović visited Amsterdam on a stopover between New York and Copenhagen. He gave a lecture at the Institute for Theoretical Physics, went to a museum, had dinner with the queen and, of course, was interviewed by Stroom.

As Cvitanović was born in Croatia. we talked about the situation there.

"I say that I was born in Croatia. because Yugoslavia doesn't exist anymore. Yesterday I brought a list of destroyed cultural monuments in Croatia, to hand it to the queen. The war in Croatia is very brutal and very ugly. Apparently they want to destroy the people and their culture. My mother is 67 and an art historian. She normally writes books. Now she's out in the countryside running around trying to save sculptures by taking them out of the churches, while they're shooting across the river. You can't imagine it, it's like a cowboy movie. This war is bad news for the whole of Europe, but the world doesn't seem to care. The only civilized thing to do would be to send troops if the fighting hasn't stopped by a certain date. Of course the situation is messy, of course it is unpleasant, but it has to be done. Imagine a house with gangsters killing all the inhabitants and the police standing outside saying 'we won't go in because it's messy'. This means that everybody in the street will have to arm themselves. This is now happening in Eastern Europe. It is a very dangerous development. You can't have a civilized community in this way."

Having finished high school, Mr. Cvitanović went to America. Before going to MIT, he had several jobs in factories. He took his PhD at Cornell in high energy physics: he computed the magnetic moment of the electron up to twelve digits. He worked at Stanford and Princeton before he changed his subject to nonlinear dynamics. "I started working on nonlinear dynamics in 1976, but nobody wanted to hear about it. The first time I could give a lecture about it was in 1982." Now he is a research professor at the Niels Bohr Institute in Copenhagen.

"Why did you choose physics?"

"First I wanted to be a fireman, like everyone else. Then I wanted to be an artist, but you can't live from that. After that I wanted to be an architect, but there are too many architects. I happened to be quite good at mathematics. So when I went to MIT. I decided to do physics, because I thought physics was good there. Like every graduate student, once a year I thought that I should change to a more sensible subject. But now I'm too old to change."

"Why did you change from high energy physics to nonlinear dynamics?"

"Even in fundamental research there is a difference between applied fundamental research and fundamental fundamental research. As a high energy physicist I worked on the purely fundamental problem of quark confinement. Now when I look back I think that I've done some decent work, as good as many others. But in retrospect we achieved nothing. When I started in high energy physics there was still some kind of interaction with the experiment. Even though the theory was very hard to understand, the magnetic moment of the electron was still a tangible thing. But later it became harder and harder to make sense of the stuff. I have great respect for my colleagues in high energy physics. Technically they are masters of what they do, but the subject is adrift. It doesn't seem to connect to anything and it makes outrageous claims, like a theory of everything, which is totally ridiculous because it does nothing whatsoever. But while I was doing high energy physics, which I was paid for, I got interested in nonlinear dynamics. I worked on this just for fun. Now I'm very happy that I did it, because it actually turned out to be much more interesting. In nonlinear dynamics there were very simple, tangible things. It was 19th century physics and mathematics you could explain to any competent person who was interested. It is a nice time because it's still easy to make progress. New tools are found, little ideas work, lots of little things can be done before the subject starts getting complicated baroque and rococo again. The curious thing is that people who worked on this subject at the time came from very different fields like meteorology, mathematics, biology and physics. These people didn't start talking to each other just because they enjoyed to cross disciplines, but because they found themselves in the same situation. Their problems were different but the mathematics to formulate them converged. I contributed for example to the development of the theory of period doubling, having no idea that it could be used for anything whatsoever. But it can actually be seen in helium and mercury heated from below in a desk top experiment.

Because nonlinear dynamics appeared in so many fields, articles about the subjects were published in many different journals, so they were hard to find. When I started giving lectures, I made a collection of the papers I considered most important. The collection was so useful, that I turned it into a book called Universality in Chaos. So in the past it turned out that the things I did just for myself paid off, while the things I was doing because I had to do them were drowned in the general frustration of thousands of colleagues doing the same thing. Now I worry because I'm only doing nonlinear dynamics. I don't have the time to do anything else secretly.

"Some people don't seem to take nonlinear dynamics very seriously. They think it's merely playing with the computer."

"This criticism is well deserved. Some people really just run computers to produce pictures. The computer is useful as a visualizing tool, but it doesn't represent any understanding. There are many charlatans in this field. But the people who understand more of nonlinear dynamics realize how difficult the real problems are. Mathematicians for example consider the Riemann conjecture a very difficult thing. But for us it comes out as a simplifying limit of our problem. The things that most people know of nonlinear dynamics are the things one can easily explain. They are of course not representative for the subject."

"What are you working on now?"

"I'm working on a thing called quantum chaos. From my point of view it is indistinguishable from classical chaos. It just turns out that the same formalism can very pleasantly be used both in classical and in quantum context. One just puts in some little i's and \hbar's. I enjoy it very much, especially as I stopped working on high energy physics because, in the seventies, I understood that even for very simple nonlinear problems we had no good picture and no good methods. And we tried to solve the strongly nonlinear theory of Quantum ChromoDynamics! It seemed outrageous that we thought we could solve a quantum field theory problem, while we didn't know how to solve it for nonlinear oscillators. I felt that one should go and develop a technique for solving the classical system you want to quantize."

"Can you tell us how the Feigenbaum-Cvitanović equation was found?"

"The story is just anecdotal. Feigenbaum is a friend of mine. Once I went to a lecture he gave. The talk wasn't very good in my opinion: there was hardly a chance that anybody could understand it. So I checked very carefully what he was doing. Working on it, I got interested and eventually I found a sort of compact summary of his work expressed in this equation. I wrote it for him only to help him to present his work. That's how the equation came about: Feigenbaum had the vision to work on this problem and I had the technical luck to write it in this way. It amuses me that now have my name on this equation. One has done many things and for some reason something gets picked out. And work you personally consider much more profound and difficult gets thrown away by everybody, including yourself eventually. You have to understand that this equation is not codified knowledge: it's not like the books of Torah, a Divine Truth written down in the only possible way. It's an attempt to write a set of phenomena in a compact way, a historical accident. I think there will be a different, broader understanding that will just throw this away and place it in another framework."

## Education

"Unfortunately students are taught as if all equations and theories are sacred. It's as though differential equations always have to be written as an x with a dot on it, there should be a Schr{\"o}dinger equation and there should be three Keplerian laws. Students are credulous, they tend to trust authority. But actually it's a dishonest thing to teach them this way. It may even be harmful, because they are robbed of the uncertainty that is necessary to do original physics. They are taught as though they were learning how to play a violin, but afterwards they are expected to compose a symphony. So when I gave a quantum theory course, I tried to start from the scratch and teach them the essence. I told them that they did not have to do the complicated commutators with thousands of indices, because this is just a technical detail that you can learn. But it turned out that what I thought to be important did not interest them. They really enjoyed putting these stupid indices! What can I say..."

"So what is your recipe for a good physicist?"

"First of all I don't think that being a physicist is something special. I know the choice of a profession is rather accidental. Very easily the same person could have ended up as an insurance agent or a physicist. Secondly the division between physics and mathematics is extremely irritating. People can do both and develop mathematics because of physics and vice versa. But then they pass some medieval guild definition, which says that the mathematician is the guy who proves theorems and the physicist the guy who publishes fifty papers a year. Of course it is hard for universities to change, because the money comes in a specific way. But you shouldn't be forced to choose a subject because some older professor tells you what to do or by the fact that it will be funded. I think it's ideal if you can change your subject, even radically, especially if I look at my own career.

You have to be able to choose your own subject, if you are a very good physicist. If you are not a very good physicist, there is no point in being a physicist at all. Science is an expensive thing for society to support. Bad physicists are useless. In fact most people are capable of doing physics. You don't have to be brilliant for it. You just have to believe in it. A good physicist wants to do physics for its own intrinsic reasons. Because he thinks it's interesting. You should not waste your life on it, if you don't care about it. If you want to do it because you want to become a professor, you should study history, or some other subject where being a professor is really a respectable occupation. Many people get pushed into science if they show any talent in mathematics. The result is that many extremely competent scientists are not scientists at all. If science means having technical knowledge or skill, then they are scientists. But physics is the philosophy of nature, so if you show no interest whatsoever in this world that physics is supposed to explain, you're not a scientist."

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## Periodic orbits and quantum chaos

"In high school you learn for example about Keplerian motion. Such a stable periodic solution is very exceptional. In nature nearly all dynamical systems are chaotic. These systems are described by nonlinear dynamics. A chaotic system also has periodic solutions. But they are unstable. So they exist only mathematically and not in nature. There are a great many of these solutions. Therefore you can build a tree of them, on which you suspend the chaotic motion. If you want to compute some average, instead of running all over the system, you take the average for the tree of periodic motions. This method also works in the context of quantum mechanics. Bohr described the electron moving around the nucleus in a similar motion to that of planets moving around the sun. The only difference is that the 'particle-planet' is replaced by a 'wave-electron'. Now nonlinear dynamics tells you that the orbit of the electron is actually chaotic. Here again you use the tree of unstable periodic orbits: you put the wave on a cloud of them. In other words you do quantization on a set of periodic orbits instead of just one."

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## Historical picture

In his talk at the institute Cvitanović drew a historical picture of the development of quantum chaos. For those who missed it, here we give a summary.

"The theory of quantum chaos is the result of three parallel developments in mathematics, classical mechanics and quantum mechanics. The people working in these fields had no idea that their subjects were related. Number theory was always assumed to be pure mathematics and to have nothing to do with applied mathematics like dynamics. It was started by Gauss and formulated by Riemann in terms of the Riemann zeta functions. In the forties however, Selberg connected the Riemann zeta functions to dynamics, but it was still artificial from the physical point of view. Classical mechanics was developed in parallel. Poincar\'e was the first to be aware of the problems of nonlinearity. After him mostly mathematicians and mathematical physicists continued the work. They applied it to celestial mechanics. It joined physics only in the seventies again by the work of Smale, Sinai and Ruelle. Then the tools we use today, like zeta functions and periodic orbits, appeared. The development of quantum mechanics has two phases. The old quantum theory started in 1900 with Planck's paper. The main contributions were by people like Bohr and Sommerfeld. In 1925, the break year, matrix mechanics was introduced and the quantum mechanics we use nowadays was formulated. One of the main reasons why the old quantum theory went into a deep crisis and which led to a totally new formulation is, while it was extremely successful in explaining the hydrogen atom, it absolutely failed in the next easiest problem: the helium atom. Many people like Bohr, Kramer, Pauli and Heisenberg worked very hard on it, but these attempts were not successful. Only a few months ago the problem has been solved by using periodic orbits. It could have been done years ago if the people working on it were aware of the possibility of chaos and non-integrability. They might have been if they had paid more attention to one of Einstein's papers. In this paper he points out that the Bohr-Sommerfeld quantization must fail if there are not enough integrals of motion and that in that case Poincar\'e had shown that the behavior is more complicated. Today we work with chaos on a skeleton of periodic orbits. It has to do with statistical mechanics and now even goes back to quantum mechanics. The mathematics we use is connected to the Riemann's zeta function. It's curious that much we do today was available fifty years or even a century ago."

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## The Feigenbaum-Cvitanović equation

By iterating the function xn+l=fA(X_n) for different values of A we obtain the following picture with periodic orbits and chaos.

In order to compute the scaling factor $\alpha$ between the distances d in the figure, Feigenbaum and Cvitanović developed their equation:

The z in this polynomial is the order of the maximum of the function f. If you put the polynomial $g(x)$ into the equation, the rescaling factor $\alpha$ comes out. The curious thing is that $\alpha$ only depends on the order of the maximum of f, and not on f itself. In nature this order is always two so $\alpha$ is universal. Its value is 2.502907....