John Guckenheimer

Professor of Mathematics and Theoretical and Applied Mechanics

Cornell University

    John Guckenheimer
    Mathematics Department, Malott Hall
    Cornell University
    Ithaca, NY 14853-2401
    (607) 255-8290
    (607) 255-7149 (fax)

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.