Competition between transients in the rate of approach to a fixed point

Abstract

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.