This award supports Michael Kirby, and several other professors and graduate students of the Department of Mathematics of Colorado State University, to collaborate in applied mathematics with Professor Werner Guettinger and others of the Institute for Theoretical Physics of the University of Tuebingen, Germany. The subject of their collaboration is the study of the complex spatio-temporal behavior of nonlinear dynamical systems with many degrees of freedom. Their goal is to reduce high dimensional evolution systems to low dimensional dynamical systems and to analyze those with the qualitative geometrical concepts of nonlinear dynamics. To do this they must develop suitable methods for mathematical approximation of invariant manifolds. The research benefits from the German group's pioneering work in developing low-dimensional dynamical systems theory and applying it to physical problems. The joint research also requires the complementary expertise of the US group in dimensionality reduction and symmetry exploitation. In particular, they have developed pattern analysis methods for dynamical systems based on neural network modeling algorithms Starting from fundamental principles of mathematics and physics, these collaborating scientists are investigating the development of organized structures and moving patterns whose activation is triggered when external forces exceed critical values. They seek to demonstrate how symmetrical or coherent spatio-temporal structures may arise, and why such structures are preferred in nature. The answers to such questions also have bearing on larger issues of technological development.
