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John Guckenheimer

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Professor of Mathematics and Theoretical and Applied Mechanics

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Cornell University

Address:
John Guckenheimer
Mathematics Department, Malott Hall
Cornell University
Ithaca, NY 14853-2401
Phone:
(607) 255-8290
(607) 255-7149 (fax)
E-mail:
gucken@cam.cornell.edu

Curriculum Vitae
Classes
Research
DsTool
Publications
Opinions

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 sytems 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 myself and former students 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 algorithm development for problems involving periodic
orbits and upon applications to the neurosciences, animal locomotion
and control of nonlinear systems.