Michigan State University

Title: Summer Experimental Mathematics

Directors: Aklilu Zeleke

Dates of Program: May 17 - June 27, 2009

Summary: The general theme of SEM is discrete dynamical systems. Such systems have been extensively used in modeling real life situations: Atmospheric changes and neural networks to mention few. We focus on one dimensional discrete dynamical systems for this year's program. Consider a one dimensional map of the form x_n₊₁=x_n-d_n+p_n where d_n and p_n account respectively for the number of cells destroyed and released into the blood stream during the time interval from n to n+1. The parameters d_n and p_n depend on x_n. There are different suggestions for the form of this dependence. One option is to let d_n=ax_n,p_n=b(x_n)^re^-sx_n,a’(0,1],b,r,s>0 to be determined. Using analytic arguments and computer simulations one can investigate the dynamics of this map and discuss issues related to the stability of the system, period doubling bifurcation and chaos (in the sense of existence of period 3 orbits and Lyapunov exponents).