Texas A&M University, Department of
Mathematics, 216 Milner Hall, 10th of October 2007, 3:00-3:50
Groups and Dynamics Seminar
Of random time
delays, rabbits, and the visible points of the integer lettuce
Tamás Kalmár-Nagy of
Texas A&M
University
The salient feature of today's ubiquitous embedded systems is the
coupling of dynamic components via the underlying communication
network. Developing a theoretical and computational framework for
stability analysis for systems with random time delays (due to the
communication) is therefore of great theoretical and practical
importance. Motivated by this need, the author started to investigate
digital control systems with random time delays. This problem gives
rise to discrete-time jump linear systems, where the transition jumps
are modeled with an underlying finite-state Markov chain. One of the
simplest, non-trivial such model is the so-called random Fibonacci
sequence. This is a second-order difference equation where the
coefficient matrix is randomly chosen (from a set of two integer valued
matrices) at every time step. In this presentation we show some
beautiful and far-reaching connections between the random Fibonacci
recurrence, the visible points of the plane, a random walk on the
induced self-similar graph and generalized Catalan numbers.