Where to place a hole to achieve fastest escape (What are the best sink and source in a network)

School of Mathematics Colloquium
Thursday, February 24, 2011 - 11:05
1 hour (actually 50 minutes)
Skiles 269
Georgia Institute of Technology
Consider any dynamical system with the phase space (set of all states) M. One gets an open dynamical system if M has a subset H (hole) such that any orbit escapes ("disappears") after hitting H. The question in the title naturally appears in dealing with some experiments in physics, in some problems in bioinformatics, in coding theory, etc. However this question  was essentially ignored in the dynamical systems theory. It occurred that it has a simple and counter intuitive answer. It also brings about a new characterization of periodic orbits in chaotic dynamical systems. Besides, a duality with Dynamical Networks allows to introduce dynamical characterization of the nodes (or edges) of Networks, which complements such static characterizations as centrality, betweenness, etc. Surprisingly this approach allows to obtain new results about such classical objects as Markov chains and introduce a hierarchy in the set of their states in regard of their ability to absorb or transmit an "information". Most of the results come from a finding that one can make finite (rather than traditional large) time predictions on behavior of dynamical systems even if they do not contain any small parameter. It looks plausible that a variety of problems in dynamical systems, probability, coding, imaging ... could be attacked now. No preliminary knowledge is required. The talk will be accessible to students.