Lyapunov quantities, limit cycles and strange behavior of trajectories in two-dimensional quadratic systems

The computation of Lyapunov quantities is closely connected with the important in engineering mechanics question of dynamical system behavior near to boundary of the stability domain. Followed by the work of Bautin [1], one differs "safe" or "dangerous" boundaries, a slight shift of which implies a small (invertible) or noninvertible changes of system status, respectively. Such changes correspond, for example, to scenario of "soft" or "hard" excitations of oscillations, considered by Andronov [2]. In classical works of Poincare [3] and Lyapunov [4] for the analysis of system behavior near boundary of the stability domain was developed the method of computation of so-called Lyapunov quantities (or Poincare-Lyapunov constants, Lyapunov coefficients, focus values), which determine a system behavior in the neighborhood of boundary. This method also permits us effectively to study the bifurcation of birth of small cycles [1, 6–15], which correspond in mechanics to small vibrations. In the present work the method of Lyapunov quantities is applied to investigation of small limit cycles. A new method for computation of Lyapunov quantities, developed for the Euclidian coordinates and in the time domain, is suggested. The general formula for computation of the third Lyapunov quantity for Lienard system is obtained. Also, the computer modeling of large (normal amplitude) limit cycles are carried out. The transformation of quadratic system to a special type of Lienard system is used for investigation of large limit cycles. For this type of Lienard systems there is obtained a domain on the plane of two parameters of system, which the systems with three small and one large cycles correspond to (around two different stationary point). In our computer experiments the effects of trajectories "flattening", that make the computational modeling difficult, are observed.