Hilbert's 16th Problem and bifurcations of Planar Polynomial Vector Fields

By Jibin Li. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2003

504 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/88ebdf59feca07e9f23a2eba6fe7ebdfe064057b

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Citations in the corpus

A Chebyshev criterion for Abelian integrals (2008, background): “Solving this problem, even in the case d = 2, seems to be out of reach at the present state of knowledge (see the works of Ilyashenko [17] and Li Jibin [20] for a survey of the recent results on the subject).”
On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line (2014, background): “For the moment this problem remains open, for more details on the 16th Hilbert’s problem see for instance [12, 16, 18].”
On the birth of limit cycles for non-smooth dynamical systems (2012, background): “study of the limit cycles of the differential systems in dimension two is one of the main problems of the qualitative theory of differential systems in dimension two, see for instance the surveys of Ilyashenko [15] and Jibin Li [17].”
On Hopf bifurcation in non-smooth planar systems (2010, background): “More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type), different from the case of smooth systems [8,11].”
Limit Cycles Near Homoclinic and Heteroclinic Loops (2008, unknown): “See [2,7,8,10].”
Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields (2005, background): “To begin with, we take the standard complex form of general Z2-equivariant vector fields, described by Li [3]: _ z 1⁄4 F 2ðz; zÞ; _ z 1⁄4 F 2ðz; zÞ; ð5:1Þ where F 2ðz; zÞ 1⁄4 P ðw1;w2Þ þ iQðw1;w2Þ; w1 1⁄4 12 ðzþ zÞ; w2 1⁄4 1 2i ðz zÞ, P, Q, w1 and w2 are all real; and F 2ðz; zÞ 1⁄4 ðA0 þ A1jzj2 þ A2”
New Results on the Study of Zq-Equivariant Planar Polynomial Vector Fields (2010, background): “In 2003, some study results around the perturbed Zq-equivariant planar polynomial Hamiltonian vector fields have been stated in our survey article [ 4 ] (and see [5–12]).”