Nests of limit cycles in quadratic systems

Abstract

Abstract We give a proof of the distribution property of limit cycles in so-called quadratic systems. We prove that the possible limit cycle distributions are either ( n , 0 ) \left(n,0) or ( n , 1 ) \left(n,1) (where n ∈ { 0 } ∪ N n\in \left\{0\right\}\cup {\mathbb{N}} ). The aim of this article is to simplify and fill gaps in the original proof by Zhang (On the distribution and number of limit cycles for quadratic systems with two foci, Qual. Theory Dyn. Sys. 3 (2002), 437–463). The sixteenth Hilbert problem asks for an upper bound for the number of limit cycles in polynomial systems H ( n ) H\left(n) , where n n is the maximum degree of the polynomial defining the system. A consequence of the distribution property is that it reduces the study of H ( 2 ) H\left(2) to the study of the maximum number of limit cycles surrounding one singularity.