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 ""safe"" or ""dangerous"" boundary of the stability domain. In classical works for the analysis of system behavior near boundary of the stability domain was developed the method of Lyapunov quantities (or Poincare-Lyapunov constants), which determine system behavior in the neighborhood of the boundary. In the present work a new method for computation of Lyapunov quantities, developed for the Euclidian coordinates and in the time domain, is suggested and is applied to investigation of small limit cycles. The general formula for computation of the third Lyapunov quantity for Lienard system is obtained. Transformations between quadratic system and special type of Lienard system are described. The computation of large (normal amplitude) limit cycles for quadratic systems such that the first and second Lyapunov quantities are equal to zero and the third one is not equal zero were carried out. In these computations the quadratic system is reduced to the Lienard equation and by the latter the two-dimensional domain of parameters, corresponding the existence of four limit cycles (three ""small"" and one ""large"") was evaluated. This domain extends the domain of parameters obtained for the quadratic system with four limit cycles due to Shi in 1980"