Singular points and limit cycles of planar polynomial vector fields

2000 ◽  
Vol 102 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Ilia Itenberg ◽  
Eugeniĭ Shustin
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Singular Points ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

Limit cycles of three-dimensional polynomial vector fields

Nonlinearity ◽  
2004 ◽  
Vol 18 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Marcin Bobie ski ◽  
Henryk o a dek
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

BIFURCATIONS OF LIMIT CYCLES IN A Z2-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELD OF DEGREE 7

2006 ◽  
Vol 16 (04) ◽  
pp. 925-943 ◽  
Author(s):  
JIBIN LI ◽  
MINGJI ZHANG ◽  
SHUMIN LI
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Vector Fields ◽  
Compound Eyes ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Planar Dynamical Systems ◽  
Equivariant Vector Field

By using the bifurcation theory of planar dynamical systems and the method of detection functions, the bifurcations of limit cycles in a Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 7 are studied. An example of a special Z2-equivariant vector field having 50 limit cycles with a configuration of compound eyes are given.

scholarly journals GENERATION OF SYMMETRIC PERIODIC ORBITS BY A HETEROCLINIC LOOP FORMED BY TWO SINGULAR POINTS AND THEIR INVARIANT MANIFOLDS OF DIMENSIONS 1 AND 2 IN ℝ3

2007 ◽  
Vol 17 (09) ◽  
pp. 3295-3302 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Main Tool ◽  
Singular Points ◽  
Dimensional Sphere ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Heteroclinic Loop

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e+ and e- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e- to e+, Γ is contained in ℝ3 and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e+ and e-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ.

scholarly journals Limit cycles of planar polynomial vector fields

Scholarpedia ◽  
2010 ◽  
Vol 5 (8) ◽  
pp. 9648 ◽  
Author(s):  
Maoan Han ◽  
Jibin Li ◽  
Chengzhi Li
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Polynomial Vector ◽  
Polynomial Vector Fields
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