  The main objective of this article is to study both dynamic and structural
transitions of the Taylor-Couette flow, using the dynamic transition theory and
geometric theory of incompressible flows developed recently by the authors. In
particular we show that as the Taylor number crosses the critical number, the
system undergoes either a continuous or a jump dynamic transition, dictated by
the sign of a computable, nondimensional parameter $R$. In addition, we show
that the new transition states have the Taylor vortex type of flow structure,
which is structurally stable.
