  We consider causal higher order theories of relativistic viscous
hydrodynamics in the limit of one-dimensional boost-invariant expansion and
study the associated dynamical attractor. We obtain evolution equations for the
inverse Reynolds number as a function of Knudsen number. The solutions of these
equations exhibit attractor behavior which we analyze in terms of Lyapunov
exponents using several different techniques. We compare the attractors of the
second-order M\"uller-Israel-Stewart (MIS), transient DNMR, and third-order
theories with the exact solution of the Boltzmann equation in the
relaxation-time approximation. It is shown that for Bjorken flow the
third-order theory provides a better approximation to the exact kinetic theory
attractor than both MIS and DNMR theories. For three different choices of the
time dependence of the shear relaxation rate we find analytical solutions for
the energy density and shear stress and use these to study the attractors
analytically. By studying these analytical solutions at both small and large
Knudsen numbers we characterize and uniquely determine the attractors and
Lyapunov exponents. While for small Knudsen numbers the approach to the
attractor is exponential, strong power-law decay of deviations from the
attractor and rapid loss of initial state memory is found even for large
Knudsen numbers. Implications for the applicability of hydrodynamics in
far-off-equilibrium situations are discussed.
