Qualitative behavior of non-Abelian Toda-like flows

In this paper we analyze and compare the long term behavior of certain non-Abelian generalizations of the integrable Toda lattice flow. In particular, we show that these flows exhibit generalized sorting behavior analogous to the sorting behavior of the original tridiagonal Toda lattice. We show that one of these generalizations, the block double bracket equation, is gradient with respect to a normal metric, and exactly solvable. This provides a new solvable case of the double bracket system.