The main mathematical goal of this proposal is the study of parameter space for polynomial diffeomorphisms in the complex domain. In particular, the PI will consider the complex Henon map. The Henon map is the simplest model of sophisticated dynamics in two higher dimension dynamics. With the aid of the holomorphic structure, complex dynamics can help bridge the separation existing between theoretical real dynamical systems and experimental dynamical systems. The proposer plans to continue studying the structure of attracting orbits. She will also study the analytical and geometrical properties of the dynamics that occur in the vicinity of certain types of bifurcations for complex maps. The dynamics close to some types of bifurcations is rich and varied. Strange attractors, infinitely many basins, and cascade of horseshoes are coexisting and they are still not well-understood. Gavosto will study these issues using a combination of analytic tools from several complex variables and immersive mathematical visualization techniques. The PI and collaborators will continue to develop computer visualization tools based on structured sets of textured surfaces. These tools allow real time interaction with complicated sets of higher dimensional data. The PI will incorporate undergraduate and high school students as research assistants in the visualization of data from research projects in complex dynamics, mathematical biological models, and crystal engineering.
The subject of dynamical systems models the evolution of phenomena that change over time. Numerous algebraic equations which have no solutions on the real numbers can be easily solved in the complex numbers. Along the same lines, the study of dynamical systems using complex numbers adds degrees of freedom (space dimensions) and permits the visualization of hidden structures. Theoretical and computational progress in dynamical systems will permit the decoding of intricate patterns and chaotic behavior in a variety of applications. The development of technology in the last decade has produced an explosion of data. The accurate and relevant visualization of large sets of data presents many challenges in the mathematical and physical sciences. New visualization tools will be very useful to help bridge the gap between disciplines. The outreach activities involving high school students and teachers in this project will provide a way to integrate the community into the development of science and in the understanding of its consequences. As used in this project, technology is an exceptional medium for introducing students early in their studies to new areas of research in math and science. For these reasons, the project is expected to have a broad multidisciplinary impact not only in the sciences but also possibly in the way the sciences are taught at the High School level and above. This project is jointly funded by the Office of Multidisciplinary Activities and the Division of Mathematical Sciences of the Directorate of Mathematical and Physical Sciences.