  Dynamical systems studies of differential equations often focus on the
behavior of solutions near critical points and on invariant manifolds, to
elucidate the organization of the associated flow. In addition, effective
methods, such as the use of Poincare maps and phase resetting curves, have been
developed for the study of periodic orbits. However, the analysis of transient
dynamics associated with solutions on their way to an attracting fixed point
has not received much rigorous attention. This paper introduces methods for the
study of such transient dynamics. In particular, we focus on the analysis of
whether one component of a solution to a system of differential equations can
overtake the corresponding component of a reference solution, given that both
solutions approach the same stable node. We call this phenomenon tolerance,
which derives from a certain biological effect. Here, we establish certain
general conditions, based on the initial conditions associated with the two
solutions and the properties of the vector field, that guarantee that tolerance
does or does not occur in two-dimensional systems. We illustrate these
conditions in particular examples, and we derive and demonstrate additional
techniques that can be used on a case by case basis to check for tolerance.
Finally, we give a full rigorous analysis of tolerance in two-dimensional
linear systems.
