  Vesicles under shear flow exhibit various dynamics: tank-treading ($tt$),
tumbling ($tb$) and vacillating-breathing ($vb$). A consistent higher order
theory reveals a direct bifurcation from $tt$ to $tb$ if $C_a\equiv \tau
\dot\gamma $ is small enough ($\tau$= vesicle relaxation time towards
equilibrium shape, $\dot\gamma$=shear rate). At larger $C_a$ the $tb$ is
preceded by the $vb$ mode. For $C_a\gg 1$ we recover the leading order original
calculation, where the $vb$ mode coexists with $tb$. The consistent calculation
reveals several quantitative discrepancies with recent works, and points to new
features. We analyse rheology and find that the effective viscosity exhibits a
minimum at $tt-tb$ and $tt-vb$ bifurcation points.
