2006/11/06 by Valery A. Gaiko, Gaiko, Valery A. · 1 citation
Mathematics · Physics and Astronomy · #34C05 #34C07 #34C23 #37G05 #37G10 #37G15 #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.CA #math.DS #msc:34C05 #msc:34C07 #msc:34C23 #msc:37G05 #msc:37G10 #msc:37G15
paper · pdf · doi:10.48550/arxiv.math/0611143
15 pages
arxiv created 2006/11/06 · openalex publication_date 2006/11/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.