Dulac - Cherkas method for detecting the exact number of limit cycles surrounding one or three equilibrium points of planar autonomous systems [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Alexandr Grin
Yanka Kupala State University of Grodno


For smooth autonomous systems the problem of precise non-local estimation of the limit cycles number in a simply-connected domain of a real phase plane containing one or three equilibrium points with a total Poincaré index +1 is considered. To solve this problem, we present new approaches that are based on a sequential two-step construction of the Dulac or Dulac-Cherkas which provide the closed transversal curves decomposing the simply-connected domain in simply-connected subdomains, doubly-connected subdomains, and possibly a three-connected subdomain. As an additional approach, we consider a generalization of the Dulac-Cherkas method, where the traditional requirement can be weakened and replaced by the condition of the transversality of the curves on which the divergence vanishes. The efficiency of the developed approaches is demonstrated by several examples of some classes of the polynomial systems, for which it is proved that there exists a limit cycle in each of the doubly-connected subdomains and two limit cycles in the three-connected subdomain.