Stable matrices in Turing pattern formation

A sqaure matrix A is a stable one if all of its eigenvalues have negative real parts, as the solutions of corresponding linear differential equation converge to zero as time goes to infinity. In 1952 Alan Turing found that for a 2*2 stable matrix A, it is possible that A-tD becomes unstable for some diagonal positive matrix D and some positive number t. This becomes the mathematiccal foundation of theory of morphogenesis pattern formation, which has inspired many recent breakthrough in biology and chemistry. We would like to consider the stability changes of the matrix A-tD for n*n matrix A and diagonal positive matrix D when t changes from zero to infinity. In particular, we could consider the case of n=3 and n=4.

Advisers: Chi-Kwong Li, Junping Shi