John Guckenheimer will carry out research in a number of different areas. These include one dimensional mappings, quadratic vector fields in the plane and multiple bifurcations. He has already established himself as an international leader in all of these areas. The current proposal is a continuation of this very successful program. Over the last decade there has been a rapid growth in the mathematical theory of iterations of one dimensional mappings. This theory has been successfully applied in such diverse areas as the theory of chemical reactors, the dynamics of semiconductor devices and the transition to chaotic behaviour in fluid dynamics. Guckenheimer has a good record of interaction with scientists in other fields. His research into multiple bifurcations will be aimed at classifying types of degenerate bifurcations which occur near quadratic Hamiltonian systems. This will be used to find solutions to a restricted form of Hilbert's 16th problem.
