Bendixson-Dulac criterion and reduction to global uniqueness in the problem of estimating the number of limit cycles

For autonomous systems on the real plane, we develop a regular method for localizing and estimating the number of limit cycles surrounding the unique singular point. The method is to divide the phase plane into annulus-shaped domains with transversal boundaries in each of which a Dulac function is constructed by solving an optimization problem, which permits one to use the Bendixson-Dulac criterion. We state the principle of reduction to global uniqueness and use it in the case of existence of an Andronov-Hopf function of limit cycles to obtain a sharp global estimate of the number of limit cycles for an individual system as well as for a one-parameter family of such systems in an unbounded domain.