The cyclicity of a cubic system with nonradical Bautin ideal

Journal of Differential Equations

Volumn 246, Issue 3, 2009, Pages 1274-1287

  Levandovskyy, Viktor ^a   Romanovski, Valery G ^b   Shafer, Douglas S ^c  

^a RWTH AACHEN UNIVERSITY   (Germany)

^b UNIVERSITY OF MARIBOR   (Slovenia)

^c UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE   (United States)

Author keywords

Bautin ideal; Cyclicity; Polynomial system

Indexed keywords

EID: 55649118439     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.07.026     Document Type: Article

References (23)

1. Adams W.W., and Loustaunau P. An Introduction to Gröbner Bases. Grad. Stud. Math. vol. 3 (1994), American Mathematical Society, Providence, RI
2. Bautin N.N. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Mat. Sb. (N.S.) 30 (1952) 181-196
3. Bautin N.N. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Trans. Amer. Math. Soc. 100 (1954) 181-196
4. Buchberger B. Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems (An Algorithmic Criterion for the Solvability of Algebraic Systems of Equations). Aequationes Math. 4 (1970) 374-383 English translation
5. Buchberger B. Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems (An Algorithmic Criterion for the Solvability of Algebraic Systems of Equations). In: Buchberger B., and Winkler F. (Eds). Groebner Bases and Applications. London Math. Soc. Lecture Note Ser. vol. 251 (1998), Cambridge Univ. Press 535-545
6. Cima A., Gasull A., Mañosa V., and Manosas F. Algebraic properties of the Liapunov and periodic constants. Rocky Mountain J. Math. 27 (1997) 471-501
7. Cox D., Little J., and O'Shea D. Ideals, Varieties, and Algorithms (1992), Springer-Verlag, New York
8. Françoise J.-P., and Yomdin Y. Bernstein inequalities and applications to analytic geometry and differential equations. J. Funct. Anal. 146 (1997) 185-205
9. Greuel G.-M., and Pfister G. A SINGULAR Introduction to Commutative Algebra (2002), Springer-Verlag, New York
10. G.-M. Greuel, G. Pfister, H. Schönemann, Singular 3.0. A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, 2005, http://www.singular.uni-kl.de
11. W. Decker, G. Pfister, H.A. Schönemann, Singular 2.0 library for computing the primary decomposition and radical of ideals primdec.lib, 2001
12. Han M., Zang H., and Zhang T. A new proof to Bautin's theorem. Chaos Solitons Fractals 31 (2007) 218-223
13. Krick T., and Logar A. An algorithm for the computation of the radical of an ideal in the ring of polynomials. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. New Orleans, LA, 1991. Lecture Notes in Comput. Sci. vol. 539 (1991), Springer-Verlag, Berlin 195-205
14. Liu Y.-R. Formulas of values of singular point and the integrability conditions for a class of cubic system, M (3) ≥ 7. Chinese Sci. Bull. 35 (1990) 1241-1245
15. Liu Y.-R., and Li J.-B. Theory of values of singular point in complex autonomous differential systems. Sci. China Ser. A 33 (1990) 10-23
16. Romanovski V.G., and Robnik M. The center and isochronicity problems for some cubic systems. J. Phys. A: Math. Gen. 34 (2001) 10267-10292
17. Romanovski V.G. Computational algebra and limit cycles bifurcations in polynomial systems. Nonlinear Phenom. Complex Syst. 10 (2007) 79-85
18. Roussarie R. Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem. Progr. Math. vol. 164 (1998), Birkhäuser, Basel
19. Sibirskii K.S. On the number of limit cycles in the neighborhood of a singular point. Differ. Uravn. 1 (1965) 53-66 (in Russian)
20. Yakovenko S. A geometric proof of the Bautin theorem. In: Il'yashenko Yu., and Yakovenko S. (Eds). Concerning the Hilbert 16th Problem. Amer. Math. Soc. Transl. Ser. 2 vol. 165 (1995), Amer. Math. Soc., Providence, RI 203-219
21. Żoła̧dek H. Quadratic systems with center and their perturbations. J. Differential Equations 109 (1994) 223-273
22. Żoła̧dek H. On a certain generalization of Bautin's theorem. Nonlinearity 7 (1994) 273-279
23. Żoła̧dek H. Eleven small limit cycles in a cubic vector field. Nonlinearity 8 (1995) 843-860