In conversation with John Guckenheimer by Hinke Osinga University of Bristol, UK

At the recent SIAM Conference on Applied Dynamical Systems John Guckenheimer, chair of the Activity Group on Dynamical Systems and former SIAM president, talked to Hinke Osinga about the role of software in dynamical systems, the growing demand for tools to investigate complex dynamical systems, and the roles of mathematics and computer science departments in supporting the development of numerical algorithms and software.

"From the beginning I was interested in real applications in dynamical systems and I recognized that the computer is a really valuable tool in doing that."

John's research has always been closely tied with computational results. However, he feels that research on dynamical systems theory involving algorithm development has often fallen into gaps between disciplines. Mathematics, computer science and applications disciplines have not fully embraced the challenges of providing the best possible computational tools for the simulation and analysis of complex systems. This was the theme of his SIAM Past Presidental address (SIAM News, Volume 32, No. 8, October 1999).

Curriculum vitae of Professor John Guckenheimer

John got his undergraduate degree in 1966 from Harvard and a PhD from Berkeley in 1970. As with many mathematicians, there is an interesting story behind how he decided to do a PhD in Berkeley. His plan was to join the Peace Corps going to Nepal and so he went to Hawaii for their training program. On his 21st birthday the last week of September he was `deselected.' "I had also thought about doing a PhD and had applied to three universities. However, I was on Hawaii and it was already early afternoon. So, only the graduate office in Berkeley had not yet gone home for the day, and they were happy to let me change my mind and accept their offer of admission."

		Before receiving his PhD form Berkeley, John spent three months in Brasil and 1.5 years at Warwick. He spent two years at the Institute of Advanced Studies in Princeton, then one year at MIT before obtaining a tenure track position in 1973 at the University of California at Santa Cruz. He stayed there until 1985, by then a Full Professor in Mathematics, and moved to Cornell University. Cornell appealed to John for their really good interdisciplinary interactions in applied mathematics. Furthermore, Cornell was just starting its Theory Center and the Mathematical Sciences Institute supported by the Army Research Office. He has been at Cornell every since.
A school picture from 2nd or 3rd grade.

John's research interests are a combination of pure dynamical systems theory, applications in (mainly) mathematical biology, and the development of algorithms for the numerical exploration of the associated models. In the first couple of years of his career he worked on the problems proposed by Smale in his 1967 Manifesto for Dynamical System Theory. He was further influenced by the research in catastrophe theory of Christopher Zeeman (at Warwick) and René Thom (whose book "Stabilité structurelle et morphogénèse: esai d'une théorie générale des modèles" was translated from French by David Fowler while he was at Warwick). At Princeton he sat in on a developmental biology seminar led by experimental biologist Malcolm Steinberg. He also attended Gordon Conferences on mathematical biology during this period and met George Oster at one of these. At the 1973 Gordon Conference, Jim Yorke talked about his now famous paper with Li, Period 3 implies Chaos. When John moved to Santa Cruz that year, he and George Oster started working together on low-dimensional population models, influenced by the experiments on blowflies done by Alexander J. Nicholson in Australia in the 1950s. Oster also worked with Bob May on one-dimensional iterations as population models; see, for example, Bifurcations and Dynamic Complexity in Simple Ecological Models, The American Naturalist 110(974): 573-599, 1976.

David Fowler, Steve Smale, John Guckenheimer, and Charles Pugh in Warwick, 1968

In those years, John was very much interested in one-dimensional maps. He met Mitchell Feigenbaum at a Gordon conference in 1975 and was blown away by his presentation. The work of Gollub, Swinney, and Libchaber on chaotic fluid flows created an enormous amount of excitement on the part of physicists in nonlinear dynamics that was reflected in meetings sponsored by the New York Academy of Sciences in 1976 and 1978. He spent the fall semester of his sabbatical in 1978-1979 at the Courant Institute of Mathematical Sciences where he met Ed Spiegel who got him involved into fluid dynamics. "Courant really was an eye-opener for me to see applied mathematics from a different perspective."

John Guckenheimer is best known for his book Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields with Phil Holmes. John met Phil for the first time in the summer of 1977 at a conference in Southampton, where he also met David Rand. Phil and David had plans for writing a book in early 1980. David visited Santa Cruz for a semester and attended an Institute of Theoretical Physics program in Santa Barbara, but he decided not to pursue the book project. John on the other hand "got tired of going around giving survey lectures on dynamical systems". Both he and Phil had lecture notes from courses they had taught, so they decided to proceed. It took them about nine to twelve months to finish their manuscript. They approached Springer-Verlag as a publisher for the manuscript. Their editor, Walter Kauffman-Bühler, was willing to take a risk in pricing the book at a lower cost than normal so that it would be affordable by individuals. Walter said that books which sold 10,000 copies were mathematical best-sellers. "We crossed that threshold quite a while ago." [Ed: Sales are now estimated at approximately 20,000 copies.]

The influence of the computer on dynamical systems

John has always used the computer as a tool to investigate the behavior of dynamical systems. While collaborating with Oster in 1974, he used the arpanet to access a mainframe at UCLA "doing things that today look ridiculously simple". During a visit to Los Alamos in 1976, Alan Perelson, who had been a postdoc of Aaron Katchalsky jointly with George Oster, helped John to investigate the Lorenz system. These investigations led to his subsequent work on the geometric structure of the Lorenz attractor. In this period, physics students Doyne Farmer, Jim Crutchfield, Norm Packard and Rob Shaw formed a group at Santa Cruz that used analog and later digital computers to study nonlinear dynamics. John Guckenheimer served as an informal advisor to this group. Thomas Bass recounts their extracurricular efforts to predict in real time where a roulette ball would stop in his book "The Eudaemonic Pie." John got his first computer in 1984 (a SUN1 whose speed was 1 kiloflops at a price of $ 30,000) and used it to run simulations of two-dimensional population models and to perform normal-form calculations on fluid models.

In the California Sierra, 1974.		John with his son Matt in 1980

John started getting serious about developing a dynamical systems tools package in 1989, after he had visited the Institute of Advanced Studies for a special year on dynamical systems. His postdoc Swan Kim had joined him in Princeton and had been working on a set of bifurcation tools for studying iterations of the two-torus. These efforts evolved into the computer package KAOS that was a precursor to DsTool. MSI supported this research with money for a SUN3. A year after Swan Kim finished KAOS, John obtained support from NSF for three years to work on computational mathematics. He used it to start the development of the dynamical systems toolbox DsTool. His student Mark Myers was the lead person in this effort. Mark brought his experience as an engineer at JPL to his graduate studies at Cornell. He was involved in the trajectory planning for the Gallileo spacecraft, among other things. Mark's experience in writing software was instrumental in making DsTool a success. Graduate students Patrick Worfolk and Rick Wicklin joined Mark in writing the initial versions of DsTool. At the end of the grant, John sought funding to continue the project, but was told that "writing software is a commercial business." Not wanting to start a commercial venture, he turned his efforts to research on algorithm development. For example, he wrote the papers Computing Hopf Bifurcations I with Mark Myers and Bernd Sturmfels, and Defining Functions for Multiple Hopf Bifurcations with Willy Govaerts and Alexander Khibnik.

Already in 1990 John's vision was that research in applied mathematics should be coupled with good applications. He found good examples for his research on computational methods for dynamical systems in two different areas: the challenges of multiple time scales in models for neuroscience, and the class of hybrid systems in control theory problems. Software development remains important in this research, but "I don't think that anybody has really figured out what the role of software is in a research enterprise." The evolution of open-source software has changed the landscape for such work, but many research communities still have difficulty in determining how to best support and recognize the development of software for their communities.

The political side of things

In 1997-1998, when John was president of SIAM, he was very concerned about the fact that in the US work on numerical analysis and numerical algorithms was falling between the cracks, with neither computer science nor mathematics wanting to take responsibility as the primary disciplinary home for these research areas. He finds it frustrating to see that this is still the case. "I find the world of computational science and engineering a very confusing landscape. There are different groups with very different agendas. Selected areas such as numerical linear algebra and optimization have developed definite niches. Some disciplines such as astrophysics, atmospheric sciences, computational fluid dynamics, computational chemistry and genomics have developed strong traditions within CSE. Some groups are oriented towards building extremely large computers and seem to measure work as much in terms of how many cycles are used as in the scientific advances accomplished with these computers."

Numerical analysis was a large component of computer science during its formative years, but computer science as a discipline has evolved away from this area. The discipline has become more focused on information storage and retrieval, the internet and computers as communication rather than computing devices. Data on PhDs in computer science published by the Computer Research Association document how few students are choosing to work in scientific computation or numerical analysis. "Numerical analysis per se is something that seems to be diminishing in computer science departments. Where does this stuff get done?" The answer is "in between the cracks". Scattered mathematics, applied mathematics and engineering departments provide homes for research on numerical algorithms, but the unmet needs for industry and science are enormous.

Industry would like to rely more heavily upon computation to design and develop their products, but the algorithmic infrastructure to support their efforts for the analysis of complex dynamical systems simulations is inadequate. "We could be much better at this given the hardware we have!" Without the proper acknowledgement and associated support, relatively few talented people get drawn into software development for dynamical systems. John thinks that the numerical analysis of dynamical systems ought to be part of SIAM's continuing efforts to strengthen computational science and engineering.

From a university perspective it is not clear what the most effective strategies might be. Should one focus upon mathematics, applied mathematics, computer science or other departments, or upon interdisciplinary programs? Of course, things differ from country to country, but in the US mathematics departments have little space for this kind of research and teaching loads are typically higher than in departments with a greater emphasis upon computational science. "Developing software in a mathematics department is similar to running a laboratory, but mathematics departments do not have the resources to support laboratory research. Experimentalists typically teach fewer formal classes than mathematicians." Computer science departments are more entrepeneurial. "If the Federal Government would make algorithm development a high priority as part of its initiatives to develop computational science, then computer scientists would start doing it again."

John can also envisage the start of new departments. He sees the development of better algorithms and software for studying increasingly complex systems as a bottleneck for the growth of computational science and engineering. "I would like to see an approach that brings more mathematics into these efforts. The ability of mathematicians to distill the essence of these problems and pursue novel approaches for their solution can make an enormous difference, in the future as it has repeatedly in the past." And then, with an eye on the research interests of his interviewer: "You would like to have a toolkit for working with manifolds as computational objects. Is it feasible? Sure it is feasible, but it takes a certain amount of organization and it will not get recognized as fundamental research until it is done and people see how great it is."

Snowbird Utah, May 2005.