Qualitative Tools for Studying Periodic Solutions and Bifurcations as Applied to the Periodically Harvested Logistic Equation

One-dimensional differential equations dy/dt = f (t, y) that are periodic in t arise in many areas of applied mathematics. The attracting and repelling periodic solutions and their bifurcations are the key qualitative features that determine the observed behavior of the system being modeled. One quickly finds that the low dimensional character of the problem conceals difficulties that in practice are resolved by numerical simulation. However, these simulations are guided by theoretical results and techniques, as suggested in a related context by Poincaré in his wonderful statement on rigor quoted above. The authors encountered such a differential equation when studying a simplified version of the Wilson-Cowan model of two periodically stimulated coupled neurons [29]. To assemble the needed theoretical toolkit, we turned to the logistic equation with periodic harvesting, a population model that we teach in our undergraduate classes and that is featured in many introductory texts [2], [3], [4]. These tools are presented below using the logistic equation as an example, and they should be of help to students and researchers in their own simulations and theoretical studies. In addition, some suggestions for further work are given in the last section. We also touch on the inspiring yet wavering history of science and mathematics in three places: the development of the logistic equation and its variants, the use of Poincaré’s first-return map in dynamics and chaos theory, and the progress and lack of it that marks the investigation of Hilbert’s still unsolved sixteenth problem. It is gratifying that, even in mathematics, humble beginnings can quickly lead to grand vistas. Though we confine ourselves to real variable techniques, we allude to results obtained with complex variables, as in the bounds on the number of periodic solutions given in a beautiful yet brief paper of Ilyashenko [17].