John GuckenheimerMathematics Department
565 Malott Hall
Ithaca, NY 14853-2401
Telephone: (607) 255-8290
Fax: (607) 255-7149
I engage in research on dynamical systems and their
applications. Even the simplest
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
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.