John Guckenheimer
Encyclopedia
John Guckenheimer joined the Department of Mathematics at Cornell University
in 1985. He was previously at the University of California at Santa Cruz (1973-1985). He was a Guggenheim fellow in 1984, and was elected president of the Society for Industrial and Applied Mathematics
in 1996. Guckenheimer received his B.A. from Harvard University
in 1966 and his Ph.D. from the University of California at Berkeley in 1970.
His Ph.D. thesis advisor was Stephen Smale
.
His book Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (with Philip Holmes
) is an extensively cited work on dynamical systems.
, algorithms for periodic orbits
, and dynamics in systems with multiple time scales.
of crustaceans - attempting to learn more about neuromodulation
, the ways in which the rhythmic output of the STG
is modified by chemical and electrical inputs.
, Guckenheimer has constructed a new family of algorithms that compute periodic orbits
directly. His research in this area attempts to automatically compute bifurcations
of periodic orbits as well as "generate rigorous computer proofs of the qualitative properties of numerically computed dynamical systems".
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
in 1985. He was previously at the University of California at Santa Cruz (1973-1985). He was a Guggenheim fellow in 1984, and was elected president of the Society for Industrial and Applied Mathematics
Society for Industrial and Applied Mathematics
The Society for Industrial and Applied Mathematics was founded by a small group of mathematicians from academia and industry who met in Philadelphia in 1951 to start an organization whose members would meet periodically to exchange ideas about the uses of mathematics in industry. This meeting led...
in 1996. Guckenheimer received his B.A. from Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
in 1966 and his Ph.D. from the University of California at Berkeley in 1970.
His Ph.D. thesis advisor was Stephen Smale
Stephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...
.
His book Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (with Philip Holmes
Philip Holmes
Philip J. Holmes is the Eugene Higgins Professor of Mechanical and Aerospace Engineering at Princeton University. As a member of the Mechanical and Aerospace Engineering department, he formerly served as the interim chair until May 2007....
) is an extensively cited work on dynamical systems.
Research
Dr. John Guckenheimer's research has focused on three areas - neuroscienceNeuroscience
Neuroscience is the scientific study of the nervous system. Traditionally, neuroscience has been seen as a branch of biology. However, it is currently an interdisciplinary science that collaborates with other fields such as chemistry, computer science, engineering, linguistics, mathematics,...
, algorithms for periodic orbits
Orbit (dynamics)
In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...
, and dynamics in systems with multiple time scales.
Neuroscience
Guckenheimer studies dynamical models of a small neural system, the stomatogastric ganglionStomatogastric nervous system
The Stomatogastric Nervous System is a commonly studied neural network composed of several ganglia in arthropods that controls the motion of the gut and foregut. The network of neurons acts as a central pattern generator. It is a model system for motor pattern generation because of the small...
of crustaceans - attempting to learn more about neuromodulation
Neuromodulation
In Neuromodulation several classes of neurotransmitters regulate diverse populations of central nervous system neurons...
, the ways in which the rhythmic output of the STG
Stomatogastric nervous system
The Stomatogastric Nervous System is a commonly studied neural network composed of several ganglia in arthropods that controls the motion of the gut and foregut. The network of neurons acts as a central pattern generator. It is a model system for motor pattern generation because of the small...
is modified by chemical and electrical inputs.
Algorithms for Periodic Orbits
Employing automatic differentiationAutomatic differentiation
In mathematics and computer algebra, automatic differentiation , sometimes alternatively called algorithmic differentiation, is a set of techniques to numerically evaluate the derivative of a function specified by a computer program...
, Guckenheimer has constructed a new family of algorithms that compute periodic orbits
Orbit (dynamics)
In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...
directly. His research in this area attempts to automatically compute bifurcations
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
of periodic orbits as well as "generate rigorous computer proofs of the qualitative properties of numerically computed dynamical systems".
Dynamics in systems with Multiple Time Scales
Guckenheimer's research in this area is aimed at "extending the qualitative theory of dynamical systems to apply to systems with multiple time scales". Examples of systems with multiple time scales include neural systems and switching controllers.DsTool
Guckenheimer's research has also included the development of computer methods used in studies of nonlinear systems. He has overseen the development of DsTool, an interactive software laboratory for the investigation of dynamical systems.Selected publications
- Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (with P. Holmes), Springer-Verlag, 1983, 453 pp.
- Phase portraits of planar vector fields: computer proofs, J. Experimental Mathematics 4 (1995), 153–164.
- An improved parameter estimation method for Hodgkin-Huxley model (with A. R. Willms, D. J. Baro and R. M. Harris-Warrick), J. Comp. Neuroscience 6 (1999), 145–168.
- Computing periodic orbits and their bifurcations with automatic differentiation (with B. Meloon), SIAM J. Sci. Stat. Comp. 22 (2000), 951–985.
- The forced van der Pol equation I: the slow flow and its bifurcations (with K. Hoffman and W. Weckesser), SIAM J. App. Dyn. Sys. 2 (2002), 1–35.
External links
- Department of Mathematics home page
- Theoretical & Applied Mechanics home page
- [ftp://cam.cornell.edu/pub/dstool DsTool FTP download area]
- Dr. John Guckenheimer user page on ScholarpediaScholarpediaScholarpedia is an English-language online wiki-based encyclopedia that uses the same MediaWiki software as Wikipedia, but has features more commonly associated with open-access online academic journals....