Some lower bounds for h(n) in Hilbert's 16th problem

Qualitative Theory of Dynamical Systems

Volumn 3, Issue 2, 2002, Pages 345-360

  Li, Jibin ^a   Chan, H S Y ^b   Chung, K W ^b  

^a KUNMING UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Distributions of limit cycles; Hilbert's 16th problem; Perturbed planar hamiltonian systems; Second bifurcation

Indexed keywords

EID: 84896693675     PISSN: 15755460     EISSN: 16623592     Source Type: Journal    
DOI: 10.1007/BF02969411     Document Type: Article

References (31)

1. M.L. Abell and J.P. Braselton, Maple V: by Example, AP Professional, Boston (1994).
2. V.I. Arnold, Geometric Methods in Theory of Ordinary Differential Equations, Springer-Verlag, New York (1983).
3. H.S.Y. Chan, K.W. Chung and Dongwen Qi, Some bifurcation diagrams for limit cycles of quadratic differential systems, Int. J. Bifurcation and Chaos 11 (2001), 197-206
4. H.S.Y. Chan, K.W. Chung and Jibin Li, Bifurcations of limit cycles in a Z3-equivariant planar vector field of degree 5, Int. J. Bifurcation and Chaos 11 (2001), 2287-2298
5. H.S.Y. Chan, K.W. Chung and Jibin Li, Bifurcations of limit cycles in Zqequivariant planar vector fields of degree 5, to appear in Proceeding of International Conference on Foundations of Computational Mathematics in honor of Professor Steve Smale's 70th Birthday, 13-17 July 2000
6. C.J. Christopher and N.G. Lloyd, Polynomial systems: lower bound for the Hilbert numbers, Proc. Royal Soc. London Ser. A 450 (1995), 219-224
7. G.B. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, N.J. (1963).
8. M. Farkas, Periodic Motion, Springer-Verlag, New York, (1994).
9. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of vector Fields, Springer-Verlag, New York, (1983).
10. D.A. Gudkov, The topology of real projective algebraic varieties, Russian Math. Surveys 29, 4 (1974), 1-79
11. D. Hilbert, Mathematical problems, In Proceeding of Symposia in Pure Mathematics 28 (1976), 1-34
12. Yu Ilyashenko, Finiteness Theorem for Limit Cycles, American Mathematical Society, Providence, RI (1991).
13. Jibin Li and Cunfu Li, Planar cubic Hamiltonian systems and distribution of limit cycles of (E3), Acta. Math.Sinica, 28, 4 (1985), 509-521
14. Jibin Li and Qimin Huang, Bifurcations of limit cycles forming compound eyes in the cubic system, Chinese Ann. of Math. 8B (1987), 391-403
15. Jibin Li and Zhengrong Liu, Bifurcation set and compound eyes in a perturbed cubic Hamiltonian system, In 'Ordinary and Delay Differential Equations', π Pitman Research Notes in Math. Series 272 (1992), Longman, England, 116-128
16. Jubin Li and Xiaohua Zhao, Rotation symmetry groups of planar Hamiltonian systems, Ann.of Diff.Eqs. 5 (1989), 25-33
17. Jibin Li and Zhengrong Liu, Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system, Publications Mathmatiques, 35 (1991), 487-506, Spain.
18. Jibin Li, H.S.Y. Chan and K.W. Chung, Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 5, to appear in Science in China (Series A).
19. Jibin Li, H.S.Y. Chan and K.W. Chung, Investigations of bifurcations of limit cycles in a Z2-equivariant planar vector field of degree 5, to appear in Int. J. Bifurcation and Chaos.
20. N.G. Lloyd, Limit cycles of polynomial systems. In New Directions in Dynamical Systems, Ed. T. Bedford and J. Swift, 40, London Mathematical Society Lecture Notes, 192-234, (1988).
21. Dingjun Luo, Xian Wang, Deming Zhu and Mouan Han, Bifurcation Theory and Methods of Dynamical Systems, World Scientific, Singapore, (1997).
22. N.T. Otrokov, On the number of limit cycles of a differential equation in a neighbourhood of a singular point (in Russian), Mat. Sb. 34 (1954),127-144
23. L.M. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York (1991).
24. L.M. Perko, Limit cycles of quadratic systems in the plane, Rocky Mountain Journal of Math. 14 (1984), 619-645
25. V.A. Rokhlin, Complex topological characteristics of real algebraic curves, Russian Math. Surveys 33 (1978), 85-98
26. S. Smale, Mathematical Problems for the Next Century, The Mathematical Intelligencer 20, 2 (1998), 7-15
27. O.Ya. Viro, Progress in topology of real algebraic varieties over the last six years, Usp.Mat.Nauk. 41, 3 (1986), 45-67
28. G. Wilson, Hilbert's sixteenth problem, Topology 17 (1978), 53-73
29. Yanqian Ye, Theory of Limit Cycles, Transl. Math. Monographs 66. Amer. Math.Soc., Providence, RI (1986).
30. Yanqian Ye, Qualitative Theory of Polynomial Differential Systems, Modern Mathematics Series, Shanghai Scientific and Technical Publishers (1995), Shanghai (in Chinese).
31. Zhifeng Zhang, Tongren Ding, Wenzao Huang and Zhenxi Dong, Qualitative Theory of Differential Equations, Transl.Math. Monographs 101. Amer. Math.Soc., Providence, RI (1992).