  Direct simulations of two-dimensional channel flow of a viscoelastic fluid
have revealed the existence of a family of Tollmien-Schlichting (TS) attractors
that is nonlinearly self-sustained by viscoelasticity [Shekar et al., J.Fluid
Mech. 893, A3 (2020)]. Here, we describe the evolution of this branch in
parameter space and its connections to the Newtonian TS attractor and to
elastoinertial turbulence (EIT). At Reynolds number $Re=3000$, there is a
solution branch with TS-wave structure but which is not connected to the
Newtonian solution branch. At fixed Weissenberg number, $Wi$ and increasing
Reynolds number from 3000-10000, this attractor goes from displaying a
striation of weak polymer stretch localized at the critical layer to an
extended sheet of very large polymer stretch. We show that this transition is
directly tied to the strength of the TS critical layer fluctuations and can be
attributed to a coil-stretch transition when the local Weissenberg number at
the hyperbolic stagnation point of the Kelvin cat's eye structure of the TS
wave exceeds $\frac{1}{2}$. At $Re=10000$, unlike $3000$, the Newtonian TS
attractor evolves continuously into the EIT state as $Wi$ is increased from
zero to about $13$. We describe how the structure of the flow and stress fields
changes, highlighting in particular a "sheet-shedding" process by which the
individual sheets associated with the critical layer structure break up to form
the layered multisheet structure characteristic of EIT.
