John Mark Guckenheimer (born 1945) joined the Department of Mathematics at

Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teach an ...

in 1985. He was previously at the

University of California, Santa Cruz The University of California, Santa Cruz (UC Santa Cruz or UCSC) is a public university, public Land-grant university, land-grant research university in Santa Cruz, California. It is one of the ten campuses in the University of California syste ...

(1973-1985). He was a

Guggenheim fellow Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...

in 1984, and was elected president of the

Society for Industrial and Applied Mathematics Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific socie ...

(SIAM), serving from 1997 to 1998. Guckenheimer received his A.B. in 1966 from

Harvard Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...

and his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...

in 1970 from

Berkeley Berkeley most often refers to: *Berkeley, California, a city in the United States **University of California, Berkeley, a public university in Berkeley, California * George Berkeley (1685–1753), Anglo-Irish philosopher Berkeley may also refer ...

, where his Ph.D.

thesis advisor A doctoral advisor (also dissertation director, dissertation advisor; or doctoral supervisor) is a member of a university faculty whose role is to guide graduate students who are candidates for a doctorate, helping them select coursework, as well ...

was

Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...

. His book ''Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields'' (with

Philip Holmes Philip John Holmes (born May 24, 1945) 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 ...

) is an extensively cited work on

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...

Research

Dr. John Guckenheimer's research has focused on three areas —

neuroscience Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, development ...

, algorithms for

periodic orbits In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynami ...

, and dynamics in systems with multiple time scales.

Neuroscience

Guckenheimer studies dynamical models of a small neural system, the

stomatogastric ganglion The stomatogastric ganglion (STG) is a much studied ganglion (collection of neurons) found in arthropods and studied extensively in decapod crustaceans. It is part of the stomatogastric nervous system. See also * Central pattern generator Ce ...

of crustaceans — attempting to learn more about

neuromodulation Neuromodulation is the physiological process by which a given neuron uses one or more chemicals to regulate diverse populations of neurons. Neuromodulators typically bind to metabotropic, G-protein coupled receptors (GPCRs) to initiate a second ...

, the ways in which the rhythmic output of the STG is modified by chemical and electrical inputs.

Algorithms for Periodic Orbits

Employing

automatic differentiation In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation, computational differentiation, auto-differentiation, or simply autodiff, is a set of techniques to evaluate the derivative of a function ...

, Guckenheimer has constructed a new family of

algorithms In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing c ...

that compute

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".

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.

Awards and honors

He became a SIAM Fellow in 2009. In 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

.List of Fellows of the American Mathematical Society
retrieved 2013-01-19. He won a

Leroy P. Steele Prize The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories. The prizes have b ...

in 2013 for his book (coauthored with

), and he gave the Moser Lecture in May 2015.

Selected publications

* ''Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields'' (with Philip Holmes), Springer-Verlag, 1983, 453 pp. * ''Phase portraits of planar vector fields: computer proofs'', Journal of 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.

Notes

References

* *

External links

John Guckenheimer - Home page

Department of Mathematics home page

Theoretical & Applied Mechanics home page
* tp://cam.cornell.edu/pub/dstool DsTool FTP download area
Dr. John Guckenheimer
user page on

Scholarpedia ''Scholarpedia'' is an English-language wiki-based online encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content in science and medicine. ''Scholarpedia'' articles are written ...

{{DEFAULTSORT:Guckenheimer, John 1945 births Living people Harvard University alumni University of California, Berkeley alumni 20th-century American mathematicians 21st-century American mathematicians Cornell University faculty Fellows of the American Academy of Arts and Sciences Fellows of the Society for Industrial and Applied Mathematics Fellows of the American Mathematical Society Presidents of the Society for Industrial and Applied Mathematics People from Baton Rouge, Louisiana Mathematicians from Louisiana American textbook writers