About behavior of trajectories of a third-order dynamical system with quadratic nonlinearities
In the introduction an object of the research is pointed - autonomous systems of the third order, right parts of which are quadratic polynomials with real coefficients. The aim of the research is qualitative picture of the behavior of the trajectories of the ordinary dynamic differential systems with quadratic right-hand sides on the integral surfaces in the phase space. In the main part of the obtained new qualitative characteristics of the behavior of trajectories of these systems, with the first integral in the form of a family of surfaces of hyperbolic type, and also found the analytical conditions for the existence of closed trajectories and limit cycles. We obtain conditions, when the equilibrium state in the phase space of this system is a closed plane curve of the second order. Found conditions for the existence of limit cycle on the integral surfaces in the phase space of this system. Studied bifurcation of a limit cycle in a change in the parameters of the system. The pattern trajectories in the vicinity of the rest point, which lies at the origin. The results obtained in the study may be used in the qualitative theory and the theory of bifurcations of three-dimensional systems and their applications.