A real autonomous quadratic system of three differential equations with an infinite number of limit cycles

Abstract

In this paper, we consider the problem of construction of real autonomous quadratic systems of three differential equations with the nonlocal existence of an infinite number of limit cycles. This means that an infinite number of limit cycles, emerging from the focus due to the Andronov – Hopf bifurcation, can exist in the phase space not only in the vicinity of the focus and not only for parameter values close to the bifurcation value. To solve this problem we use the method of determination of limit cycles as the curves of intersection of an invariant plane with a family of invariant elliptic paraboloids. Then the study of the limit cycles of the constructed system of the third order is reduced to the study of the corresponding system of the second order on each of the invariant elliptic paraboloids. The proof of the nonlocal existence of the limit cycle and the investigation of its stability for such a second-order system is carried out by constructing a topographic system of Poincaré functions or by transforming to polar coordinates.