### Citations

702 |
Geometrical Methods in the Theory of Ordinary Differential Equations
- Arnold
- 1982
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Citation Context ...mprehensive review on the study of Hilbert’s 16th problem can be found in a survey article [2]. To help understand and attack the problem the so called weak Hilbert’s 16th problem was posed by Arnold =-=[3]-=-. The problem is to ask for the maximal number of isolated zeros of the Abelian integral or Melnikov function: Z Mðh; dÞ Q n dx Pn dy; ð2Þ Hðx;yÞh where H(x,y), Pn and Qn are all real polynomials o... |

99 |
Mathematical problems,
- Hilbert
- 1902
(Show Context)
Citation Context ...results H2(2) = 2 and H2(3) = 5, we shall show that H2ðnÞ 4 ðn þ 1Þ for 3 n =3,4,...,20. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction The study of the well-known Hilbert’s 16th problem =-=[1]-=- is still very active though it has not been completely solved even for quadratic systems, since it has great impact on the development of modern mathematics. Consider the following planar system: _x ... |

59 | 1998] ”Computation of normal forms via a perturbation technique
- Yu
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Citation Context ...luated at the origin are a purely imaginary pair, ±i. Further, suppose the normal form associated with this Hopf singularity is given in polar coordinates (obtained by using, say, the method given in =-=[9]-=-): _r r ðv0 þ v1 r 2 þ v 2 r 4 þ þv k r 2k þ Þ; ð4Þ _h 1 þ t1 r 2 þ t2 r 4 þ þtk r 2k þ ; ð5Þ where r and h represent, respectively, the amplitude and phase of the limit cycles, and vi, i =0,1,2,.... |

43 | Hilbert’s 16th problem and bifurcations of planar polynomial vector field
- Li
- 2003
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Citation Context ...ind the upper bound, called Hilbert number H(n), on the number of limit cycles that system (1) can have. A comprehensive review on the study of Hilbert’s 16th problem can be found in a survey article =-=[2]-=-. To help understand and attack the problem the so called weak Hilbert’s 16th problem was posed by Arnold [3]. The problem is to ask for the maximal number of isolated zeros of the Abelian integral or... |

35 |
The infinitesimal 16th Hilbert problem in the quadratic case,
- Gavrilov
- 2001
(Show Context)
Citation Context ...iterature, mainly published since the 90’s of last century. The conclusion is: quadratic Hamiltonian systems with 2nd-degree polynomial perturbation can have maximal two limit cycles, i.e., H2(2) = 2 =-=[12,13]-=-, where the subscript 2 denotes second-order Hamiltonian systems. Recently, it has been shown that quadratic Hamiltonian systems with 3rd-degree polynomial perturbation can have maximal five limit cyc... |

22 | Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system - Li, Liu - 1991 |

20 |
A concrete example of the existence of four limit cycles for plane quadratic systems
- Shi
- 1980
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Citation Context ...tin [5] proved that there exist 3 small limit cycles around a fine focus point or a center. Until the end of 1970’s, concrete examples are given to show that quadratic systems can have 4 limit cycles =-=[10,11]-=-, which have (3,1) configuration: three limit cycles enclose a fine focus point, while one limit cycle encloses another element focus point. In this paper, attention is focused on bifurcation of limit... |

6 |
Hopf bifurcations for near-Hamiltonian systems,”
- Han, Yang, et al.
- 2009
(Show Context)
Citation Context ...iltonian systems. Recently, it has been shown that quadratic Hamiltonian systems with 3rd-degree polynomial perturbation can have maximal five limit cycles in the vicinity of a center, i.e., H2(3) = 5=-=[14]-=-. In[14], the Melnikov function method is used, on the basis of the following Melnikov function: I MðhÞ L h q n dx Z Z pn dy DðhÞ @pn @x þ @qn @y dxdy; ð7Þ where Lh is a contour around the origin,... |

5 |
On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or centre type
- NN
- 1954
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Citation Context ...s such that v0 = v1 = v2 = = vk 1 = 0, but vk – 0, and then perform appropriate small perturbations to prove the existence of k limit cycles. For general quadratic system (1) (n = 2), in 1952, Bautin =-=[5]-=- proved that there exist 3 small limit cycles around a fine focus point or a center. Until the end of 1970’s, concrete examples are given to show that quadratic systems can have 4 limit cycles [10,11]... |

4 | Necessary and sufficient conditions for the existence of center. Dokl Akad Nauk 1944;42:160–3 - IS |

4 | Criteria for center of a differential equation. Volg Matem Sbornik 1964;2:87–91 - KE |

3 | Four limit cycles from perturbing quadratic integrable systems by quadratic polynomials
- Yu, Han
(Show Context)
Citation Context ...Hamiltonian system: dx dt y þ a1 xyþ a2 y2 ; dy dt x þ a3 2 1 x 2 a1 y2 ; with 1 ; 0 being a a3 center if a1 < a3; saddle point if a1 > a3: The details for deriving (12) from (10) can be found in =-=[15]-=-. Since we are interested in the limit cycles bifurcating from the center (0,0), we will ignore whether the singular point is a center or a saddle point. The Hamiltonian of system (12) is given by 1 ;... |

2 |
Bifurcation of limit cycles of planar systems
- Han
- 2006
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Citation Context ... of x and y with degH = n + 1, and max{degPn, deg Qn} 6 n. The weak Hilbert’s 16th problem itself is a very important and interesting problem, closely related to the following near-Hamiltonian system =-=[4]-=-: _x Hyðx; yÞþep nðx; yÞ; _y Hxðx; yÞþeq nðx; yÞ; ð3Þ where H(x,y), pn(x,y) and qn(x,y) are all polynomial functions of x and y, and 0 < e 1 is a small perturbation. Studying the bifurcation of li... |

1 |
The relative position, and the number, of limit cycles of a quadratic differential system
- LS, MS
- 1979
(Show Context)
Citation Context ...tin [5] proved that there exist 3 small limit cycles around a fine focus point or a center. Until the end of 1970’s, concrete examples are given to show that quadratic systems can have 4 limit cycles =-=[10,11]-=-, which have (3,1) configuration: three limit cycles enclose a fine focus point, while one limit cycle encloses another element focus point. In this paper, attention is focused on bifurcation of limit... |

1 |
Iliev ID. On the number of limit cycles in perturbations of quadratic Hamiltonian systems
- Horozov
- 1994
(Show Context)
Citation Context ...iterature, mainly published since the 90’s of last century. The conclusion is: quadratic Hamiltonian systems with 2nd-degree polynomial perturbation can have maximal two limit cycles, i.e., H2(2) = 2 =-=[12,13]-=-, where the subscript 2 denotes second-order Hamiltonian systems. Recently, it has been shown that quadratic Hamiltonian systems with 3rd-degree polynomial perturbation can have maximal five limit cyc... |