Abelian Integrals: From the Tangential 16th Hilbert Problem to the Spherical Pendulum

2016 ◽  
pp. 327-346
Author(s):  
Pavao Mardešić ◽  
Dominique Sugny ◽  
Léo Van Damme
Keyword(s):  
Hilbert Problem ◽  
Abelian Integrals ◽  
Spherical Pendulum ◽  
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

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.

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.

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

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.

scholarly journals Modern symbolic computation methods: Lyapunov quantities and 16th Hilbert problem

2014 ◽  
Vol 1 (16) ◽  
pp. 5
Author(s):  
Gennady A. Leonov ◽  
Nikolay V. Kuznetsov ◽  
Elena V. Kudryashova ◽  
Olga A. Kuznetsova
Keyword(s):  
Symbolic Computation ◽  
Hilbert Problem ◽  
16Th Hilbert Problem
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