Liénard's system and Smale's problem

Abstract

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 Liénard'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 Liénard'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 Liénard's system applying the Wintner-Perko termination principle for multiple limit cycles and discuss some other results concerning this system.