Dulac-Cherkas Test for Determining the Exact Number of Limit Cycles of Autonomous Systems on the Cylinder

Abstract

For real autonomous two-dimensional systems of differential equations with continuously differentiable right-hand sides 2π-periodic in one of the variables, we consider the problem of determining the exact number of limit cycles of the second kind on the cylinder. If the system has no equilibria, then we propose to solve this problem by two methods based on a successive two-step application of the Dulac-Cherkas test or one of its modifications, which permits determining closed transversal curves dividing the cylinder into subdomains surrounding it and such that the system has precisely one limit cycle of the second kind in each of them. The efficiency of the proposed methods is illustrated with examples of systems corresponding to the Abel equation, for which the number of limit cycles on the whole phase cylinder is established.