Explicit non-algebraic limit cycles for polynomial systems

2005/05/23 by Armengol Gasull, Gasull, Armengol, Hector Giacomini +3
Mathematics · #34C-05 34C-07 (Primary) 34C25 37C27 (Secondary) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CA #math.DS #msc:34C-05 #msc:34C-07 #msc:34C25 #msc:37C27

paper · pdf · doi:10.48550/arxiv.math/0505464

arxiv created 2005/05/23 · arxiv updated 2009/12/01

Abstract

We consider a system of the form x'=P_n(x,y)+xR_m(x,y), y'=Q_n(x,y)+yR_m(x,y), where P_n(x,y), Q_n(x,y) and R_m(x,y) are homogeneous polynomials of degrees n, n and m, respectively, with n<=m. We prove that this system has at most one limit cycle and that when it exists it can be explicitly found. Then we study a particular case, with n=3 and m=4. We prove that this quintic polynomial system has an explicit limit cycle which is not algebraic. To our knowledge, there are no such type of examples in the literature. The method that we introduce to prove that this limit cycle is not algebraic can be also used to detect algebraic solutions for other families of polynomial vector fields or for probing the absence of such type of solutions.

Related

• Limit cycles for planar semi-quasi-homogeneous polynomial vector fields
• Number of limit cycles for planar systems with invariant algebraic curves
• The maximal number of limit cycles in a family of polynomial systems
• The general Liénard polynomial system
• A new approach for the study of limit cycles