On the Limit Cycles of a Perturbed Z3-Equivariant Planar Quintic Vector Field

2015 ◽  
Vol 25 (05) ◽  
pp. 1550073
Author(s):  
Yunlei Ma ◽  
Yuhai Wu
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Hilbert Problem ◽  
Polynomial System ◽  
Heteroclinic Loop ◽  
Planar Vector ◽  
16Th Hilbert Problem

In this paper, the number and distributions of limit cycles in a Z3-equivariant quintic planar polynomial system are studied. 24 limit cycles with two different configurations are shown in this quintic planar vector field by combining the methods of double homoclinic loops bifurcation, heteroclinic loop bifurcation and Poincaré–Bendixson Theorem. The results obtained are useful to the study of weakened 16th Hilbert problem.

ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES IN A QUINTIC PLANAR VECTOR FIELD

2008 ◽  
Vol 18 (07) ◽  
pp. 1939-1955 ◽  
Author(s):  
YUHAI WU ◽  
YONGXI GAO ◽  
MAOAN HAN
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Hilbert Problem ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Analysis Method ◽  
Quintic Polynomial ◽  
Hilbert Number ◽  
Planar Vector ◽  
16Th Hilbert Problem

This paper is concerned with the number and distributions of limit cycles in a Z2-equivariant quintic planar vector field. By applying qualitative analysis method of differential equation, we find that 28 limit cycles with four different configurations appear in this special planar polynomial system. It is concluded that H(5) ≥ 28 = 52+ 3, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to the study of the second part of 16th Hilbert problem.

scholarly journals On the 16th Hilbert problem for algebraic limit cycles

2010 ◽  
Vol 248 (6) ◽  
pp. 1401-1409 ◽  
Author(s):  
Jaume Llibre ◽  
Rafael Ramírez ◽  
Natalia Sadovskaia
Keyword(s):  
Limit Cycles ◽  
Hilbert Problem ◽  
16Th Hilbert Problem ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

scholarly journals Limit Cycles Coming from Some Uniform Isochronous Centers

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
Haihua Liang ◽  
Jaume Llibre ◽  
Joan Torregrosa
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Hilbert Problem ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
16Th Hilbert Problem ◽  
Isochronous Centers

AbstractThis article is about the weak 16th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centersof degree

scholarly journals On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

2011 ◽  
Vol 250 (2) ◽  
pp. 983-999 ◽  
Author(s):  
Jaume Llibre ◽  
Rafael Ramírez ◽  
Natalia Sadovskaia
Keyword(s):  
Limit Cycles ◽  
Algebraic Curves ◽  
Hilbert Problem ◽  
16Th Hilbert Problem

Bifurcation of Limit Cycles in a 12-Degree Hamiltonian System Under Thirteenth-Order Perturbation

2018 ◽  
Vol 28 (01) ◽  
pp. 1850011
Author(s):  
Jianping Shi ◽  
Wentao Jiang
Keyword(s):  
Vector Field ◽  
Hamiltonian System ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Order Perturbation ◽  
Hamiltonian Vector Field ◽  
Bifurcation Of Limit Cycles ◽  
Detection Function ◽  
System A ◽  
Planar Vector

This paper considers the weakened Hilbert’s 16th problem for symmetric planar perturbed polynomial Hamiltonian system. A [Formula: see text]-equivariant planar vector field of degree 12 is deduced to find as many as possible limit cycles and their configuration patterns. By using bifurcation theory of planar dynamical system and the method of detection function, we have obtained that, under the thirteenth-order perturbation, the above [Formula: see text]-equivariant planar perturbed Hamiltonian vector field of 12-degree has at least 117 limit cycles. Moreover, this paper also shows the configuration of compound eyes of the corresponding perturbed systems.

TWELVE LIMIT CYCLES IN A CUBIC CASE OF THE 16th HILBERT PROBLEM

2005 ◽  
Vol 15 (07) ◽  
pp. 2191-2205 ◽  
Author(s):  
P. YU ◽  
M. HAN
Keyword(s):  
Limit Cycles ◽  
Multiple Scales ◽  
Hilbert Problem ◽  
Perturbation Technique ◽  
Local Limit ◽  
Polynomial Equations ◽  
Computationally Efficient ◽  
Degree Polynomial ◽  
Planar System ◽  
16Th Hilbert Problem

In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in literature for a cubic order planar system is eleven limit cycles. The system considered in this paper has a saddle point at the origin and two focus points which are symmetric about the origin. This system was studied by the authors and shown to exhibit ten small limit cycles: five around each of the focus points. It will be proved in this paper that the system can have twelve small limit cycles. The major tasks involved in the proof are to compute the focus values and solve coupled enormous large polynomial equations. A computationally efficient perturbation technique based on multiple scales is employed to calculate the focus values. Moreover, the focus values are perturbed to show that the system can exactly have twelve small limit cycles.

On the limit cycles of a quintic planar vector field

2007 ◽  
Vol 50 (7) ◽  
pp. 925-940 ◽  
Author(s):  
Yu-hai Wu ◽  
Li-xin Tian ◽  
Mao-an Han
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Planar Vector

Seven Limit Cycles Around a Focus Point in a Simple Three-Dimensional Quadratic Vector Field

2014 ◽  
Vol 24 (06) ◽  
pp. 1450083 ◽  
Author(s):  
Yun Tian ◽  
Pei Yu
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Vector Fields ◽  
Center Manifold ◽  
Three Dimensional ◽  
Recursive Formula ◽  
Polynomial Systems ◽  
Computationally Efficient ◽  
Planar Vector

In this paper, we show that a simple three-dimensional quadratic vector field can have at least seven small-amplitude limit cycles, bifurcating from a Hopf critical point. This result is surprisingly higher than the Bautin's result for quadratic planar vector fields which can only have three small-amplitude limit cycles bifurcating from an elementary focus or an elementary center. The methods used in this paper include computing focus values, and solving multivariate polynomial systems using modular regular chains. In order to obtain higher-order focus values for nonplanar dynamical systems, computationally efficient approaches combined with center manifold computation must be adopted. A recently developed explicit, recursive formula and Maple program for computing the normal form and center manifold of general n-dimensional systems is applied to compute the focus values of the three-dimensional vector field.

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