John Guckenheimer
Mathematics Department565 Malott Hall
Cornell University
Ithaca, NY 14853-2401
Telephone: (607) 255-8290
Fax: (607) 255-7149
E-mail: jmg16@cornell.edu
Research Overview
I have engaged in research on dynamical systems and their applications for close to forty years. 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 and formulas that describe their behavior seldom exist. Computer simulation is one 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 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, turbulent combustion and control of nonlinear systems.