  In 2015, Bailleul presented a mechanism to solve rough differential equations
by constructing flows, using the log-ODE method. We extend this notion in two
ways: On the one hand, we localize Bailleul's notion of an almost-flow to solve
RDEs on manifolds. On the other hand, we extend his results to non-geometric
rough paths, living in any connected, cocommutative, graded Hopf algebra. This
requires a new concept, which we call a pseudo bialgebra map. We further
connect our results to Curry et al (2020), who solved planarly branched RDEs on
homogeneous spaces.
