A study on the existence of limit cycles of A planar system with third-degree polynomials

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 14, Issue 1, 2004, Pages 41-60

  Han, M ^a   Lin, Y ^a   Yu, P ^b  

^a SHANGHAI JIAO TONG UNIVERSITY   (China)

^b UNIVERSITY OF WESTERN ONTARIO   (Canada)

Author keywords

Center; Focus value; Hilbert's 16th problem; Limit cycle; Normal form; Planar system

Indexed keywords

DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; MATHEMATICAL MODELS; PROBLEM SOLVING;

FOCUS VALUES; HILBERT'S 16TH PROBLEM; LIMIT CYCLES; NORMAL FORMS; PLANAR SYSTEMS;

POLYNOMIALS;

EID: 2942646689     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127404009247     Document Type: Article

References (20)

1. Bautin, N. N. [1952] "On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type," Mat. Sbornik (N.S.) 30, 181-196.
2. i to have a center," Diff. Urav. 14, 1594-1600.
3. Chow, S.-N., Li, C.-Z. & Wang, D. [1994] Normal Forms and Bifurcation of Planar Vector Fields (Cambridge University Press, Cambridge).
4. Christopher, C. J. & Lloyd, N. G. [1990] "On the paper of Jin and Wang concerning the conditions for a center in certain cubic systems," Bull. London Math. Soc. 22, 5-12.
5. Farr, W. W., Li, C. Z., Labouriau, I. S. & Langford, W. F. [1989] "Degenerate Hopf bifurcation formulas and Hubert's 16th problem," SIAM J. Math. Anal. 20, 13-30.
6. Jin, X. & Wang, D. [1990] "On the conditions of Kukles for the existence of a center," Bull. London Math. Soc. 22, 1-4.
7. Kukles, I. S. [1944] "Necessary and sufficient conditions for the existence of center," Dokl. Akad. Nauk 42, 160-163.
8. Li, J. & Liu, Z. [1991] "Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system," Publ. Math. 35, 487-506.
9. Liu, Y. [1987] "Formula of focal value, condition of center and central for a class of planar cubic system," Chinese Sci. Bull. 2, 85-87.
10. Liu, Y. & Li, J. [1989] "On the singularity values of complex autonomous differential systems," Sci. China (Series A) 3, 245-255 (in Chinese).
11. Lloyd, N. G. & Pearson, J. M. [1992] "Computing center conditions for certain cubic systems," J. Comput. Appl. Math. 40, 323-336.
12. Lunkevich, V. A. & Sibirsky, K. S. [1965] "Conditions of a center in homogeneous non-linearities of third degree," Diff. Eqs. 1, 1164-1168 (in Russian).
13. Malkin, K. E. [1964] "Criteria for center of a differential equation," Volg. Matem. Sbornik 2, 87-91.
14. Qin, Y. X. & Liu, Z. Q. [1981] "Computer deduction of formulas for differential equations," Sci. Bull. 26, 388-391.
15. Shi, S. A. [1980] "A concrete example of the existence of four limit cycles for planar quadratic system," Sci. Sinica. Ser. A23, 153-158.
16. Wang, D. [1999] "Polynomial systems from certain differential equations," J. Symb. Comput. 28, 303-315.
17. Ye, W. & Ye, Y. [2001] "Qualitative theory of the Kukles systems," Ann. Diff. Eqs. 17, 275-286.
18. Yu, P. [1998] "Computation of normal forms via a perturbation technique," J. Sound Vibr. 211, 19-38.
19. Yu, P. [2001] "Symbolic computation of normal form for resonant double Hopf bifurcations using multiple time scales," J. Sound Vibr. 247, 615-632.
20. Yu, P., Zhang, W. & Bi, Q. [2001] "Vibration analysis on a thin plate with the aid of computation of normal forms," Int. J. Non-Lin. Mech. 36, 597-627.