We present a new mathematical perspective to study the dynamics of constitutive models with non-monotonic curves that naturally explains thixotropic yield stress behavior. To illustrate, a viscoelastic constitutive model, which generates a non-monotone shear stress as a function of shear rate in a steady homogeneous parallel shear flow, is investigated for the dynamics initiated by a step-up or step-down in prescribed shear stress. The stress tensor for the model is a combination of the partially extending strand convection model modified to allow the shear stress to approach a non-negative limit for large shear rates, and a Newtonian solvent contribution. We address the case where the relaxation time is large. In this limit, the first maximum in the non-monotone curve occurs at a small shear rate, characterized by a parameter epsilon << 1. There is no presumption of a yield stress, but nevertheless, we obtain yield stress behavior in this limit. Complex behaviors such as yield stress hysteresis, dependence of yield stress on time scales, thixotropy, apparent unyielding at a small non-negative shear stress, and an apparent viscosity which evolves on a slow time scale, are explained by this model. The direct numerical simulation of the full governing equations is performed in conjunction with a perturbation analysis with multiple time scales, in order to characterize yielded states. A novel time-periodic fracture-heal solution, with each period composed of a short yielded flow and a long unyielded state is found. Oscillations of this nature have been reported for a soft-glassy material which fractures on a fast time scale and reheals on a slow time scale. (C) 2012 Elsevier B.V. All rights reserved,