On knife-edge conditions with unbounded growth

Imagine an economic process modeled with help of ordinary differential equations with some fixed initial conditions and suppose that a solution to the system under consideration, for some reason, should enjoy desired conditions. It appears that small changes in the system describing the modeled process may have devastating effect on the fulfillment of the desired conditions by solutions of the perturbed system. Such a feature of a system is called a knife-edge condition in the literature on economic growth and it has recently found some deeper interest in that literature. However, the available results seem to be unclear and not correct this paper shows how to correct and improve these results, creating consistent mathematical foundations of that theory. The main contribution of the paper is that knife-edge conditions are present whenever long-term unbounded growth of an economic variable is required as (or implied by) a desired regularity condition. (C) 2015 Elsevier Inc. All rights reserved.