Computation of Lyapunov quantities and limit cycles.*

The present work is devoted to the qualitative methods of study of periodic solutions of Lienard equation. This equation is often used in automatic control theory to describe the behavior of controllable generators (Cesari (1959); Lefschetz (1957)), where important tasks are to study the excitation of oscillations and analysis of limit cycles (Andronov et al. (1966); Bautin (1949)). Here, one of the most effective investigation methods is a method of computation and analysis of Lyapunov quantities (Poincare (1885); Lyapunov (1892)). In the present work the methods of computation of Lyapunov quantities and localization of limit cycles are demonstrated. These methods are applied to investigation of Lienard equation with small and large limit cycles. The expressions for the first four Lyapunov quantities for general Lienard system are obtained. By the transformation of quadratic system to a special type of Lienard equation and the method of asymptotical integration, quadratic systems with large limit cycles are investigated. The domain of parameters of quadratic systems, for which four limit cycles can be obtained, is determined.