Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem

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Wang, X. Huang and J.

Reyn, investigating various particular cases under different conditions. The final proof for general quadratic systems was given by Zhang in []. Then in , L. Chen and M. Wang [7] and S. In T. Tung [99] found out some important properties of quadratic systems: a closed orbit is convex; there is a unique singularity in the interior of it; two closed orbits are similarly resp.

Hence, the distribution of limit cycles of quadratic systems have only one or two nests. Around P. Zhang [,] proved that in two nests case at least one nest contains a unique limit cycle. Therefore, 2,2 -distribution of limit cycles for a quadratic system is impossible. Zhang and S. Cai [] and D.

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Zhu [] respectively proved that a quadratic system with a second or third order weak saddle point does not have a limit cycle. Zhang [] further proved that a quadratic system with a first order saddle point has at most one limit cycle. The global geometry of quadratic systems is also an interesting topic, for studies in this aspect see D. Schlomiuk [94,95,96], for example. There are many more other results on the non-existence, uniqueness of limit cycles, existence of two limit cycles and global phase analysis.

See Ye [] and Reyn [91]. In , Bautin [3] studied the Hopf bifurcation for quadratic systems and proved that there are at most 3 limit cycles bifurcated from a weak focus or center of such a system, and 3 limit cycles can appear. Zhu [], Joyal and Rousseau [63] and Cai and Guo [5] respectively proved that in quadratic systems there are at most 3 limit cycles bifurcated from an isolated homoclinic loop.

Concerning the Hilbert 16th Problem

However, it is still open if the number 3 can be achieved. Then Iliev [56] proved that the maximum number of limit cycles near a homoclinic loop of a degenerate Hamiltonian system is 2 under small perturbation with a single parameter by using up to the fourth order Melnikov functions. Han, Ye and Zhu [47] further proved that this conclusion is also true for the case of degenerate Hamiltonian systems under small perturbation with arbitrary parameters. Thus, in quadratic systems the maximum number of limit cycles near a homoclinic loop of any Hamiltonian system is 2 under arbitrary small perturbations.

There are also some studies on the number of limit cycles near a homoclinic loop of an integrable non-Hamiltonian system under quadratic perturbations, see He and Li [52], Han [32]. The results obtained positively suggest that 2 is also the maximum number for the integrable case.

There may appear a polycycle or heteroclinic loop with two or three elementary saddles in quadratic systems.

The results obtained by Dumortier, Roussarie and Rousseau [24], Zoladek [] and Han-Yang [45] show that at most resp, three two limit cycles can be bifurcated from a polycycle with two resp. Also, in the both cases, two limit cycles can appear. It seems very difficult even impossible to find an example of quadratic systems having three limit cycles near a polycycle with 3 saddles.


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We consider polynomial plane systems with some symmetry. Then J. Li and X. Takens [98] and V. Arnold [1].


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  • Concerning the Hilbert 16th Problem!

Han [35] and W. Li [85] Theorem 3. Recently, Han etc. For example, one can give an explicit condition which ensures the existence of 7 limit cycles near a double homoclinic loop, see Han and Zhang [49]. There are three main aspects in studying the number of limit cycles. The first is Hopf bifurcation. There have been many results in this aspect.

The main technique is to compute focus values or Liapunov constants, see for example [28] and []. By this method, P. Yu and M. Li and Y. Li, C. Liu and J. Yang [66] in the same year, using Abelian integral and [21].


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Another method is to compute the first coefficients appearing in the expansion of the first order Melnikov function at a center, say, it is proved in [46] that a center of Hamiltonian quadratic system generates at most 5 limit cycles under perturbations of cubic polynomials. The second aspect is limit cycle bifurcation from a family of periodic orbits for near-Hamiltonian or near-integrable systems. This method was first introduced by J. Li etc. See [73] for more details. By this detection function method J. Li and Q.

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There are many papers concerning the weak Hilbert's 16th problem, by using the Picard-Fuchs equations, the Argument Principle, the averaging method, the Picard-Lefschetz formula and by using some techniques, see for example [9,10,17,11,22,25,26,30,55,58,59,67,68, 69,70,71,,] etc.. The third aspect is limit cycle bifurcation near a homoclinic loop or a poly-cycle for near-Hamiltonian or near-integrable systems.

One way is to use the expansion of the Melnikov function at the loop, discussing the number of limit cycles near the loop. The method was originated by Roussarie [92] for the case of homoclinic loop and developed in [62,44,40,] for the cases of double homoclinic loops and poly-cycles. The other way is to produce limit cycles by changing the stability of a homoclinic loop originated by Han [33] and developed with many applications to polynomial systems in [47,37,45,39,49,51,50,41,42,,,29,,,,,43] etc..

In , B. There are many studies on the nonexistence and uniqueness of limit cycles and the existence of one or more limit cycles. For details, see [] and [].

Limit cycle - Encyclopedia of Mathematics

There are different methods for proving the uniqueness of limit cycles. One of them is to compare the integrals of the divergence of the system along two limit cycles, which was originated by Zhang Zhifen in see []. She proved the following which is called Zhang Zhifen's theorem:. Arnold, Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Funct.

Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of center type, Mathem. Sbornik N.

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