  We study the evolution of homogeneous Ricci solitons under the bracket flow,
a dynamical system on the space of all homogeneous spaces of dimension n with a
q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous
manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a
multiple of the identity plus a derivation) are precisely the fixed points of
the system, and that a homogeneous Ricci soliton is isometric to an algebraic
soliton if and only if the corresponding bracket flow solution is not chaotic,
in the sense that its omega-limit set consists of a single point. We also
geometrically characterize algebraic solitons among homogeneous Ricci solitons
as those for which the Ricci flow solution is simultaneously diagonalizable.
