J. Guckenheimer

John Guckenheimer

Last September was marked by the sixtieth birthday of John Guckenheimer—a professor of Mathematics at Cornell University widely renowned for his pioneering research in pure and applied dynamical systems.

John is a past president of the Society for Industrial and Applied Mathematics (SIAM), the current chair of the SIAM Activity Group on Dynamical Systems, a Guggenheim Fellow, a Fellow of the American Association for the Advancement of Science (AAAS), and an editor of many mathematical journals (including MMJ).

For many years Guckenheimer's research has been a blend of theoretical investigation, development of computer methods and studies of nonlinear systems that arise in diverse fields of science and engineering. His current work focuses upon the dynamics of systems with multiple time scales, algorithms for approximating periodic orbits and invariant manifolds, and upon applications to the neurosciences and animal locomotion.

Born and raised in Baton Rouge (Louisiana), he received a bachelor's degree in mathematics form Harvard in 1966 and went on to graduate study at Berkeley, where his interest in dynamical systems was fostered in the seminar led by Smale, Hirsch, and Pugh. His advisor at Berkeley was Stephen Smale.

Upon moving to University of California at Santa Cruz in 1973, Guckenheimer started a fruitful collaboration with George Oster (a developmental biologist and former engineering professor at Berkeley). Their joint work concerned population models and host-parasite interactions. While considering these models, Guckenheimer returned to studying the dynamics of one-dimensional maps. In a remarkable series of papers (1977–79) he demonstrated the universality of periods of stable periodic orbits and the Hölder-continuity of entropy for a wide class of unimodal smooth maps defined on the unit interval.

One of the highlights of this period was his explanation, together with Williams, of the mysterious chaotic properties of the Lorenz attractor discovered numerically by Lorenz in early 60's (a parallel explanation was found by Afraimovich, Bykov and Shilnikov).

His book with Philip Holmes on “Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields,” 1983, was the first treatise in the modern theory of dynamical systems. It remains one of the main sources in the subject, and is widely recognized for shaping the general perspective of a new generation of mathematicians and practitioners interested in applied nonlinear dynamics. The book has been updated and reprinted five times, and a Russian Translation appeared in 2004.

In 1985 John moved to Cornell University, where he assumed the Directorship of the Center for Applied Mathematics (CAM) from 1989–97. He did much to reshape CAM, and its growth and success as an interdisciplinary graduate program through the late 1980s and 1990s are largely due to his efforts.

John has long recognized the importance of systems with multiple time scales in applied mathematics. His research in this area concerns a forced van der Pol equation, which has played an important role in the history of dynamical systems. In the 1940's and 1950's, Cartwright and Littlewood showed that this system had very complicated solutions (today we would call them chaotic). Their work and that of Levinson inspired Smale's development of the horseshoe map, which has become one of the primary mathematical structures used to analyze chaotic systems. In collaboration with a group of current and former post-docs and students, Guckenheimer explained how a horseshoe map can occur in the smooth periodically forced van der Pol equation. It appears that horseshoes are generated by the canard solutions related to folded saddles on the slow surface. In earlier work, joint with Ilyashenko, he discovered a new type of canards on a two-torus. His ongoing research in this area is focused on the bifurcations of relaxation oscillations. Applications include neural and other biological models.

An important part of John's work at Cornell stems from his long collaboration with the neurobiological laboratory led by Ron Harris-Warrick. Their projects have focused on constructing dynamical models of motor neural networks in the lobster nervous system and the mouse spinal cord. These models have been used to predict useful experimental manipulations of parameters in order to understand how neurons generate rhythmic behaviors.

John also participates in an interdisciplinary multi-university research program on insect locomotion, with the dual goals of understanding biological mechanisms and building bio-inspired legged robots. In parallel, John is actively involved in the effort to educate the next generation of mathematically-savvy biologists. Together with his Cornell colleague Stephen Ellner, he designed a popular course on dynamic models in biology. Their joint textbook on the subject will appear shortly.

John's work highlights the necessary interaction between theory and computational experiments in nonlinear dynamics. In his papers, numerical experiments are often used to discover the qualitative properties of the system, which are then proven by a blend of analytic techniques and validated computations. On the other hand, in many of his projects the discovered dynamical phenomena occur in very special parameter regions and thus would be quite hard to find by naive/random numerical experimentation alone. His style of computer exploration may be compared to a series of well-planned physical experiments inspired by deep insights in theoretical physics.

John was among the pioneers of computer generated proofs in dynamical systems. He developed methods for rigorous/validated computation of phase portraits and bifurcation diagrams of multi-parametric vector fields. Together with his students and collaborators, Guckenheimer created DsTool, a widely used toolbox for interactive investigation of dynamical systems. His other contributions to numerical analysis include algorithms for approximating invariant manifolds of vector fields as well as high-order accurate methods for approximating periodic orbits.

As a member of various boards and committees, John has helped to shape and organize the activities of numerous scientific institutions including the Mathematical Sciences Research Institute (Berkeley, CA), the Institute for Mathematics and its Applications (Minneapolis, MN), the Los Alamos National Laboratory, and the European Science Foundation Network on Nonlinear Science. From 1999 till 2002 he has served as a Founding Chair of SIAM Activity Group on Life Sciences. He has been recently appointed the Associate Dean of Computing and Information Science at Cornell and is actively working to coordinate university-wide programs for training applied mathematicians, numerical analysts, and specialists in scientific computing.

Throughout his career John has advised, supervised and mentored (both formally and informally) a large number of graduate students and postdocs. Aside from his obvious scientific influence, many of his students and collaborators have been affected by John's generosity, his warm and friendly personal style, his unpretentious, level-headed, and even-handed approach both to scientific exploration and the world in general.

John is a good friend of the Russian mathematical community. During the last five years he was a Principal Co-Investigator (together with Yu. Ilyashenko) of a CRDF grant that supported a large team working in dynamical systems in Russia.

We are delighted to congratulate John on his anniversary and to wish him many more years of enjoyable and productive mathematical life!

R. Harris-Warrick, K. Hoffman, P. Holmes, Yu. Ilyashenko,
A. Khibnik, M. Tsfasman, A. Vladimirsky, W. Weckesser

MMJ Cover

Moscow Mathematical Journal
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Independent University of Moscow

Online ISSN 1609-4514
© 2005, Independent University of Moscow

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