Hilbert's 16th Problem for Quadratic Systems: New Methods Based on a Transformation to the Lienard equation

Within last century the study of cycles of twodimensional quadratic systems was stimulated by 16th Hilbert’s problem and its different variants [Hilbert, 1902; Lloyd, 1988; Blows & Perko, 1994; Ilyashenko, 2002; Chen & Wang, 1979; Shi, 1980]. The development of asymptotical methods for investigation of cycles of quadratic systems is facilitated by the fractionally-quadratic transformations, which reduce any two-dimensional quadratic system to the Lienard equation of special type with discontinuous nonlinear functions [Leonov, 1997; Leonov, 1998; Leonov, 2006]. In the present paper these new facilities are demonstrated. The elements of direct Lyapunov method, the small perturbations methods, and the bifurcation theory together permits us to obtain the existence criteria of small cycles in the neighborhoods of stationary points of Lienard equation. This criteria applied to quadratic systems. The question arises as to whether in two–dimensional quadratic systems a simultaneous bifurcation of cycles from two weak stable and weak unstable equilibria is possible by variation of one scalar parameter. Here we provide the positive answer to this question. For the Lienard equation, the classical theorems on the existence of cycles is well known [Lefschetz, 1957; Cesari, 1959]. Here there occurs quite naturally the possibility of generalization of these results and of applying them to the problem on the existence of cycles in quadratic systems. On this way in the space of parameters of quadratic systems, the sets of positive Lebesgue measure, for which the cycles exist, are discovered.