Global algebraic Poincaré-Bendixson annulus for the limit cycle of one class of a perturbed hamiltonian system
In the introduction, the object of the research is indicated - a perturbed hamiltonian system, the right parts of which are polynomials with respect to the phase variable y. The coefficients of these polynomials smoothly depend on the phase variable x and depend continuously on the parameter µ. At a zero value of the parameter, the system under consideration is the hamiltonian system. The subject of the research is the limit cycle bifurcating under the perturbation of the hamiltonian system. The aim of the study is to identify systems that have unique limit cycle in the entire phase plane for all real values of the parameter µ through analytical construction of two closed algebraic curves that form the Poincaré-Bendixson annulus. In the main part, based on the approach of L. A. Cherkas to the application of Dulac method to solve the problem of estimating the number and localization of limit cycles in the phase plane of autonomous systems of the second order, a method is proposed for constructing Dulac-Cherkas functions in the form of a polynomial of degree 2n with respect to the phase variable y, the coefficients of which smoothly depend on the second phase variable x and continuouslydepend on the parameter µ. The finding of the Dulac-Cherkas function is reduced to proving the positivity of the auxiliary polynomial. To achieve the formulated aim, a global Poincaré-Bendixon annulus is constructed, where the inner boundary is a closed curve of the zero level of the Dulac-Cherkas function, which means that there is no more than one limit cycle in the entire phase plane. The outer boundary of the ring is constructed in a special way. The results obtained can be applied in the qualitative theory and the theory of bifurcations of ordinary differential equations, as well as in the theory of nonlinear oscillations.