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John Guckenheimer

Mathematics Department               
565 Malott Hall                              
Cornell University                           
Ithaca, NY 14853-2401

Telephone: (607) 255-8290

Fax: (607) 255-7149
E-mail: jmg16@cornell.edu

Research Overview

I engage in research on dynamical systems and their applications. Even the simplest  nonlinear dynamical systems can generate phenomena of bewildering complexity. Formulas that describe their trajectories seldom exist, so computer simulations are invaluable in understanding their behavior.  Theoretical advances have been inspired by common patterns observed while simulating many different systems. One of the main goals of my research is to discover these patterns and characterize their properties. The resulting theory then serves as a guide in studying the dynamics of specific systems. It is also the foundation for numerical algorithms that seek to analyze system behavior in ways that go beyond simulation.

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. Two of the primary themes have been bifurcation theory, which  studies the  dependence of dynamical behavior upon system parameters, and the effects of multiple time scales in shaping dynamical behavior. Application areas in which I have worked include population biology, fluid dynamics, neurosciences, animal locomotion and control of nonlinear systems. My work on  algorithm development includes contributions to methods for computing bifurcations, periodic orbits and invariant manifolds of vector fields and for the analysis of fractal dimensions of attractors.