A Note on the Number of Limit Cycles in Certain Two-Dimensional Systems

Abstract

This note is concerned with certain two-dimensional differential systems x = X(x,y), y = Y{x,y). (1.1) When investigating such a system it is often desirable to be able to estimate the number of limit cycles (that is, isolated periodic orbits) in certain regions of the plane, or encircling a known critical point. The existence of a limit cycle is usually proved by using some form of the Poincare-Bendixson theorem ([5; p. 232], for example). There is, on the other hand, a variety of possible ways of showing that a system has no limit cycles or no limit cycles in certain regions. Index number arguments are often used, for example. Another common technique is to use the Bendixson criterion [5; p. 238], which is the particular case k = 0 of Theorem 1 below. Another special case of Theorem 1, this time with k — 1, is a result of Dulac [4], which in suitable circumstances yields the conclusion that (1.1) has a unique limit cycle. In the main part of this note we discuss another condition, of a different kind, which also leads to this conclusion; this result is motivated by a paper of Allwright [1]. For completeness and comparison, |we first state and prove the following result which contains as particular cases the theorems of Bendixson and Dulac to which we have referred. The idea of the proof is seen to be similar to that used by Dilberto in Theorem 7 of [3]. THEOREM 1. Suppose that D is an open, connected subset of the plane and that X and Y have continuous first order partial derivatives in D. Suppose further that there is a continuously differentiable function f: D-> 1R such that