Georgia Institute of Technology College of Sciences

It is well known that typically equations do not have analytic (expressed by formulas) solutions. Therefore a classical approach to the analysis of dynamical systems (from abstract areas of Math, e.g. the Number theory to Applied Math.) is to study their asymptotic (when an independent variable, "time", tends to infinity) behavior. Recently, quite surprisingly, it was demonstrated a possibility to study rigorously (at least some) interesting finite time properties of dynamical systems. Most of already obtained results are surprising, although rigorously proven. Possible PhD topics range from understanding these (already proven!) surprises and finding (and proving) new ones to numerical investigation of some systems/models in various areas of Math and applications, notably for dynamical analysis of dynamical networks. I'll present some visual examples, formulate some results and explain them (when I know how).