Mathematical Sciences: Computation and Analysis of Invariant Manifolds and Their Bifurcations

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In many areas of the applied sciences one is led to study dynamical systems depending on one or more parameters. The state vector often approaches a low-dimensional invariant manifold that determines the dynamical behavior, except for an initial transient. The investigator develops and analyszes methods for the direct numerical computation of such invariant manifolds in a path-following process. Also, we studies how to detect and classify potential bifurcations. So far, well-developed algorithms for path following and for detection of bifurcations exist only when the invariant manifold is a fixed point or a periodic orbit. The next interesting cases are branches of invariant 2-tori, corresponding typically to quasiperiodic motion with two frequencies. In previous work the investigator and his collaborators have developed a reliable code to compute such branches of 2-tori, but the numerical study of their bifurcations has just begun. This project extends the previous work in several directions, the most important being the following. First, indicators are added to the code that can tell if a bifurcation is possible, and -- ideally -- can tell the nature of the bifurcation. Second, to represent the invariant manifold, local charts are used instead of one global chart. This extension makes local mesh refinement practical and greatly enhances the performance and applicability of the existent code. During the last four centuries scientists have developed equations to determine the time evolution of a great variety of systems. Newton wrote down the equations for planetary motion. Euler, Navier, and Stokes developed equations to predict the flow of water and other fluids. The equations governing meteorology, oceanography, or the phase changes of materials were developed more recently. Though the equations determine the evolution in principle, supercomputers are usually necessary to actually evaluate the predictions. The reason is, of course, the great complexity of th e processes involved. In fact, even the fastest computers are often insufficient for a full modeling. Then it is necessary to ask for sensible simplifications. These are possible if the state vector (which describes the state of the system at a given instant of time) settles to a low-dimensional object in state space as time progresses. It is one aim of the proposed research to understand when such a behavior occurs. There are many known examples, ranging from applied mechanics to chemistry to material sciences. Another aim is the direct numerical computation of the low-dimensional geometric object that is approached by the state vector. The project helps understand dynamical systems better. In turn, this leads to better algorithms for predicting evolutions and makes a wider range of processes and phenomena amenable to modeling by supercomputers.