Li\'{e}nard's system and Smale's problem

In this paper, using geometric properties of the field rotation parameters, we present a solution of Smale's Thirteenth Problem on the maximum number of limit cycles for Li\'{e}nard's polynomial system. We also generalize the obtained result and present a solution of Hilbert's Sixteenth Problem on the maximum number of limit cycles surrounding a singular point for an arbitrary polynomial system. Besides, we consider a generalized Li\'{e}nard's cubic system with three finite singularities, for which the developed geometric approach can complete its global qualitative analysis: in particular, it easily solves the problem on the maximum number of limit cycles in their different distribution. We give also an alternative proof of the main theorem for the generalized Li\'{e}nard's system applying the Wintner-Perko termination principle for multiple limit cycles and discuss some other results concerning this system.