Global bifurcations of limit and separatrix cycles in a generalized Liénard system

We construct a canonical cubic dynamical system of Kukles type and carry out the global qualitative analysis of its special case corresponding to a generalized Liénard equation. In particular, we prove that the foci of such a Liénard system can be at most of second order, and that this particular system can have at least three limit cycles in the whole phase plane. Moreover, unlike all previous works on the Kukles-type systems, we study global bifurcations of limit and separatrix cycles, using arbitrary (including as large as possible) field-rotation parameters of our canonical system. As a result, we have obtained a classification of all possible types of separatrix cycles for this generalized Liénard system and also all possible distributions of its limit cycles.