Method for determining the cyclicity of singular points for quadratic systems of differential equations

Abstract

In the theory of nonlinear dynamic systems, which are described using differential equations with polynomial right-hand sides, the identification of hidden periodic regimes (so-called limit cycles) is still relevant. These periodic dynamic trajectories emerge in the vicinity of equilibrium points in a phenomenal way. The most important problem in the study of Hopf bifurcation is the search for the maximum number of limit cycles that can arise from the equilibrium point with small perturbations of the parameters of the system under study. We have studied in detail a special case of the Andronova’s system, which has two equilibrium points of the focus type, and have proven that three limit cycles arise around each of these foci. To determine the maximum limit cycle multiplicity for the system of two differential equations containing polynomials of no higher than the second degree, the values of the first three Lyapunov focal quantities were calculated. The obtained result confirms the conclusions about the degree of limit cycle multiplicity equal to three for a system of two differential equations with quadratic nonlinearity.