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| John M. Guckenheimer |
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| Professor of Mathematics |
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Web Site Contact Information
Courses & Office Hours |
Education
Ph.D. (1970) University of California at Berkeley
Research Area: Dynamical systems
Dynamical systems theory studies long time behavior of systems governed by deterministic rules. Even the simplest nonlinear dynamical systems can generate phenomena of bewildering complexity. Formulas that describe the behavior of a system seldom exist. Computer simulation is the way to see how initial conditions evolve for particular systems. In carrying out simulations with many, many different systems, common patterns are observed repeatedly. One of the main goals of dynamical systems theory is to discover these patterns and characterize their properties. The theory can then be used as a basis for description and interpretation of the dynamics of specific systems. It can also be used as the foundation for numerical algorithms that seek to analyze system behavior in ways that go beyond simulation. Throughout the theory, dependence of dynamical behavior upon system parameters has been an important topic. Bifurcation theory is the part of dynamical systems theory that systematically studies how systems change with varying parameters.
My research is a blend of theoretical investigation, development of computer methods and studies of nonlinear systems that arise in diverse fields of science and engineering. Much of the emphasis is upon studying bifurcations. The computer package DsTool is a product of the research of former students and myself with additional contributions from postdoctoral associates. It provides an efficient interface for the simulation of dynamical models and incorporates several additional algorithms for the analysis of dynamical systems. The program is freely available, subject to copyright restrictions. My current work focuses upon the dynamics of systems with multiple time scales, algorithm development for problems involving periodic orbits and upon applications to the neurosciences, animal locomotion and control of nonlinear 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.
Last modified: December 16, 2005