On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves
type: Item
instance of: scholarly article
title: On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves
zbMATH Open document ID: 1269.34038
author: Jaume Llibre
author: Natalia Sadovskaia
author: R. Ramirez
published in: Journal of Differential Equations
publication date: 2011-01-18T00:00:00Z
review text: Consider the planar polynomial system \[ {dx\over dt}= P(x,y),\quad {dy\over dt}= Q(x,y)\tag{\(*\)} \] of degree \(n\). Suppose that \((*)\) exactly \(k\) invariant algebraic curves \(g_m(x,y)= 0\), \(1\leq m\leq k\). Denote by \(A(k,n)\) the maximum number of algebraic limit cycles of \((*)\). The authors prove the following results. If the invariant algebraic curves \(g_m(x,y)= 0\), \(m= 1,\dotsc, k\), of \((*)\) are nonsingular and irreducible, then \[ A(k,n)\leq k\Biggl({n(n-1)\over 2}+ 1\Biggr)\leq {n^4\over 4}+ {3n^2\over 4}+ 1. \] If the curves \(g_m(x,y)= 0\), \(m= 1,\dotsc, k\), are irreducible and \(\sum^k_{m=1} \deg g_m\leq n+1\), then \[ A(k,n)\leq \begin{cases} {n(n-1)\over 2}+ 1\quad &\text{if }n\text{ is even},\\ {n(n-1)\over 2}\quad &\text{if }n\text{ is odd}.\end{cases} \] Additionally, sufficient conditions are provided that all algebraic limit cycles are hyperbolic.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 5838456
zbMATH Keywords: limit cycles
zbMATH Keywords: algebraic limit cycles
zbMATH Keywords: polynomial vector fields
zbMATH Keywords: 16th Hilbert problem
MaRDI profile type: Publication
full work available at URL: j.jde.2010.06.009
OpenAlex ID: W2057719662
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DOI: 10.1016/J.JDE.2010.06.009
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description: scientific article; zbMATH DE number 5838456