Renormalization and Global Bifurcation

Ibm Thomas J Watson Research Center, Yorktown Heights NY

Investigators

Abstract

The project is a continuation of DMS-9704867 which was entitled Renormalization and Global Bifurcation. Tresser and co-worker P.Coullet introduced the notion of renormalization in the theory of dynamical systems and conjectured universality of the transition to chaos in a variety of natural systems. The corresponding renormalization theory deals with the small scale geometrical structure of one-dimensional dynamical systems. Tresser used renormalization concepts to describe the bifurcation diagram of a large class of families of dynamical systems. The renormalization phenomena are observed in a wide range of applications. One would like to have a renormalization theory which is widely applicable. For This reason Martens initiated a real one-dimensional renormalization theory for unimodal maps. This work uses as less as possible underlying structure. The aim of this project is to locate more examples of renormalization, explore the corresponding bifurcation phenomena and continue the development of the above mentioned real one-dimensional renormalization theory.