The transition to turbulence in linearly stable shear flows is one of the most intriguing and outstanding problems in classical physics. It is of fundamental interest from a mathematical and physical perspective, since understanding the mechanisms that trigger turbulence would give great insight into the nature of turbulence and would provide a foundation for control of these flows. Turbulent dynamics are readily observed at flow speeds where the laminar state remains stable under infinitesimal perturbations. Moreover, for a smaller class of shear flows, such as plane Couette flow and pipe flow, linear stability theory predicts that the laminar state remains stable for all Reynolds numbers. However, numerical simulations and experiments provide evidence that these flows exhibit turbulence for sufficiently high Reynolds numbers and perturbations.

The accepted representation of the behaviors in state space postulated that the stable laminar solution coexisted with the turbulent regime. Only recently, the notion of a third, dynamically invariant region that might lie between the laminar and turbulent regions in state space has been suggested. In it, the dynamics would differ from those observed in the laminar and turbulent regimes. This boundary, called the edge of chaos, contains invariant solutions, the edge states which are too weak to become turbulent and too strong to decay to the laminar state. The edge of chaos separates trajectories that directly decay to the laminar state from those that grow and become turbulent.

These edge states, which can be either dynamically simple or complex structures, are identified using an iterated edge tracking algorithm based on a bisection method. This dissertation focuses on a dynamical systems analysis of the transition to turbulence in sinusoidal shear flow and plane Couette flow. For sinusoidal shear flow, the edge of chaos is characterized for a low-dimensional model derived via a Galerkin projection onto physically meaningful modes. The edge coincides with the codimension-1 stable manifold of an unstable periodic orbit. For the related system of plane Couette flow, direct numerical simulations of the Navier-Stokes equations are performed to identify edge states for different flow domains. For a particular range of flow geometries, multiple, non-symmetry related edge states, which coexist in state space, were found. The characterization of the edge of chaos will provide a greater understanding of the transition to turbulence in turbulent shear flows.