John Guckenheimer

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John Mark Guckenheimer
John Guckenheimer.jpg
Born 1945
Baton Rouge, Louisiana
Residence U.S.
Nationality American
Fields Mathematician
Institutions University of California, Santa Cruz
Cornell University
Alma mater Harvard University
Doctoral advisor Stephen Smale
Doctoral students Allan Willms
Known for Dynamical systems
Bifurcation theory
Notable awards Leroy P. Steele Prize (2013)

John Mark Guckenheimer (born 1945) 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 (SIAM) and served as president 1997-1998.[1] Guckenheimer received his B.A. from Harvard University in 1966 and his Ph.D. from the University of California at Berkeley in 1970.[2] His Ph.D. thesis advisor was Stephen Smale.[3]

His book Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (with Philip Holmes) is an extensively cited work on dynamical systems.

Research[edit]

Dr. John Guckenheimer's research has focused on three areas - neuroscience, algorithms for periodic orbits, and dynamics in systems with multiple time scales.[4]

Neuroscience[edit]

Guckenheimer studies dynamical models of a small neural system, the stomatogastric ganglion 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.

Algorithms for Periodic Orbits[edit]

Employing automatic differentiation, 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".

Dynamics in systems with Multiple Time Scales[edit]

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[edit]

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.[5]

Awards and honors[edit]

He became a SIAM Fellow in 2009.[6] In 2012 he became a fellow of the American Mathematical Society.[7] He won a Leroy P. Steele Prize in 2013 for his book (coauthored with Phil Holmes), and he will give the Moser Lecture in May 2015.

Selected publications[edit]

  • 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.

Notes[edit]

  1. ^ SIAM Presidents http://www.siam.org/about/more/presidents.php
  2. ^ John Guckenheimer. "John Guckenheimer, Professor of Mathematics and Theoretical and Applied Mechanics". Cornell University. Archived from the original on 2008-04-20. Retrieved 2008-03-31. 
  3. ^ The Mathematics Genealogy Project. "Mathematics Genealogy Project". North Dakota State University, Fargo, North Dakota. Retrieved 2008-03-31. 
  4. ^ John Guckenheimer. "Research". Cornell University. Archived from the original on 2008-04-20. Retrieved 2008-03-31. 
  5. ^ John Guckenheimer. "DsTool". Cornell University. Archived from the original on 2007-10-17. Retrieved 2008-03-31. 
  6. ^ SIAM Fellows http://fellows.siam.org/index.php?sort=last
  7. ^ List of Fellows of the American Mathematical Society, retrieved 2013-01-19.

References[edit]

External links[edit]