Dulac–Cherkas criterion for exact estimation of the number of limit cycles of autonomous systems on a plane

Abstract

The problem of exact nonlocal estimation of the number of limit cycles surrounding one point of rest in a simply connected domain of the real phase space is considered for autonomous systems of differential equations with continuously differentiable right-hand sides. Three approaches to solving this problem are proposed that are based on sequential two-step usage of the Dulac–Cherkas criterion, which makes it possible to find closed transversal curves dividing the connected domain in doubly connected subdomains that surround the point of rest, with the system having precisely one limit cycle in each of them. The effectiveness of these approaches is exemplified with polynomial Liènard systems, a generalized van der Pol system, and a perturbed Hamiltonian system. For some systems, the derived estimate holds true in the entire phase space.