On the discretization of double-bracket flows

This paper extends the method of Magnus series to Lie-algebraic equations originating in double-bracket flows. We show that the solution of the isospectral flow $Y'=[[Y,N],Y]$, $Y(0)=Y_0\in\Sym(n)$, can be represented in the form $Y(t)=\CC{e}^{\Omega(t)} Y_0 \CC{e}^{-\Omega(t)}$, where the Taylor expansion of $\Omega$ can be constructed explicitly, term-by-term, identifying individual expansion terms with certain rooted trees with bicolour leaves. This approach is extended to other Lie-algebraic equations that can be appropriately expressed in terms of a finite `alphabet'.