scholarly journals Uniqueness of limit cycles for quadratic vector fields

2019 ◽  
Vol 39 (1) ◽  
pp. 483-502
Author(s):  
José Luis Bravo ◽  
◽  
Manuel Fernández ◽  
Ignacio Ojeda ◽  
Fernando Sánchez
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Quadratic Vector Fields ◽  
Uniqueness Of Limit Cycles

Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

2018 ◽  
Vol 28 (11) ◽  
pp. 1850139 ◽  
Author(s):  
Laigang Guo ◽  
Pei Yu ◽  
Yufu Chen
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Center Manifold ◽  
Three Dimensional ◽  
Center Manifold Theory ◽  
Normal Form Theory ◽  
Quadratic Vector Fields ◽  
Fine Focus ◽  
Form Theory ◽  
Manifold Theory

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

Nonexistence of limit cycles for a class of structurally stable quadratic vector fields

2007 ◽  
Vol 17 (2) ◽  
pp. 259-270 ◽  
Author(s):  
J. C. Artés ◽  
◽  
Jaume Llibre ◽  
J. C. Medrado ◽  
◽  
...  
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Quadratic Vector Fields ◽  
Structurally Stable

scholarly journals Uniqueness of limit cycles in polynomial systems with algebraic invariants

1994 ◽  
Vol 49 (1) ◽  
pp. 7-20 ◽  
Author(s):  
André Zegeling ◽  
Robert E. Kooij
Keyword(s):  
Simple Proof ◽  
Limit Cycles ◽  
Real Line ◽  
Vector Fields ◽  
Polynomial Systems ◽  
Quadratic Systems ◽  
Codimension Two ◽  
Algebraic Invariants ◽  
Uniqueness Of Limit Cycles ◽  
Cubic Systems

The uniqueness of limit cycles is proved for quadratic systems with an invariant parabola and for cubic systems with four real line invariants. Also a new, simple proof is given of the uniqueness of limit cycles occurring in unfoldings of certain vector fields with codimension two singularities.

Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center

2019 ◽  
Vol 29 (03) ◽  
pp. 1950035 ◽  
Author(s):  
Jihua Wang ◽  
Yanfei Dai
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Vector Fields ◽  
Parameter Family ◽  
Reversible Systems ◽  
Abelian Integrals ◽  
Quadratic Systems ◽  
Quadratic Vector Fields ◽  
Period Annulus ◽  
Straight Line

This paper is concerned with the quadratic perturbations from one parameter family of generic reversible quadratic vector fields having a simple center and an invariant straight line. It is shown that the system can generate at least two limit cycles. As the parameter is rational, we propose a procedure for finding the upper bound to cyclicity of period annulus based on the Chebyshev criterion for Abelian integrals together with one rationalizing transformation. To illustrate our approach, we determine the cyclicity of three representative reversible systems. Our results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev [1998].

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