Limit cycle bifurcations from a non-degenerate center

Applied Mathematics and Computation

Volumn 218, Issue 9, 2012, Pages 4703-4709

  Giné, Jaume ^a  

^a UNIVERSITY OF LLEIDA   (Spain)

Author keywords

Center problem; Groebner basis; Limit cycles; Poincar Liapunov constants

Indexed keywords

CENTER PROBLEM; CERTAIN HYPOTHESIS; COMPUTATIONAL PROBLEM; CYCLICITY; GROEBNER BASIS; HOMOGENEOUS POLYNOMIALS; LIMIT CYCLE; LIMIT CYCLE BIFURCATION; POINCARE; POLYNOMIAL SYSTEMS; SINGULAR POINTS;

HOPF BIFURCATION; POLYNOMIALS;

DELTA SIGMA MODULATION;

EID: 83555172640     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2010.09.025     Document Type: Article

References (39)

1. M. Agrawal, N. Kayal, and N. Saxena Primes is in P Ann. Math. 160 2 2004 781 793
2. N.N. Bautin On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type Mat. Sb. 30 72 1952 181 196
3. N.N. Bautin On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type Amer. Math. Soc. Transl. 100 1954 397 413
4. T.R. Blows, and N.G. Lloyd The number of limit cycles of certain polynomial differential equations Proc. Roy. Soc. Edinburgh Sect. A 98 3-4 1984 215 239
5. J. Chavarriga, and J. Giné Integrability of a linear center perturbed by fourth degree homogeneous polynomial Publ. Mat. 40 1 1996 21 39
6. J. Chavarriga, and J. Giné Integrability of a linear center perturbed by a fifth degree homogeneous polynomial Publ. Mat. 41 2 1997 335 356
7. J. Chavarriga, J. Giné, and M. Grau Integrable systems via polynomial inverse integrating factors Bull. Sci. Math. 126 4 2002 315 331
8. C.J. Christopher Estimating limit cycle bifurcations from centers Differential Equations with Symbolic Computation Trends Math. 2005 Birkhuser Basel pp. 23-35
9. C.J. Christopher, and N.G. Lloyd Polynomial systems: a lower bound for the Hilbert numbers Proc. Roy. Soc. London Ser. A 45 1938 1995 219 224
10. D. Cox, J. Little, and D. O'Shea Ideals, Varieties and Algorithms second ed. Undergraduate Texts Math. 1992 Springer-Verlag
11. H. Dulac Détermination et intégration d'une certain classe d'équations différentielle ayant pour point singulier un center Bull. Sci. Math. Sér. 32 2 1908 230 252
12. M. Frommer Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmheitsstellen Math. Ann. 109 1934 395 424
13. W. Kapteyn On the midpoints of integral curves of differential equations of the first degree Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland 19 1911 1446 1457 Dutch
14. W. Kapteyn New investigations on the midpoints of integrals of differential equations of the first degree Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 20 1912 1354 1365
15. W. Kapteyn New investigations on the midpoints of integrals of differential equations of the first degree Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 21 1912 27 33 Dutch
16. W.W. Farr, C. Li, I.S. Labouriau, and W.F. Langford Degenerate Hopf-bifurcation formulas and Hilbert's 16th problem SIAM J. Math. Anal. 20 1989 13 29
17. J.P. Franoise Successive derivatives of a first return map, application to the study of quadratic vector fields Ergodic Theor. Dyn. Syst. 16 1996 87 96
18. J. Giné Polynomial first integrals via the Poincaré series J. Comput. Appl. Math. 184 2 2005 428 441
19. J. Giné The center problem for a linear center perturbed by homogeneous polynomials Acta Math. Sin. 22 6 2006 1613 1620
20. J. Giné On the number of algebraically independent Poincaré-Liapunov constants Appl. Math. Comput. 188 2 2007 1870 1877
21. J. Giné, and J. Mallol Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials Nonlinear Anal. 71 12 2009 e132 e137
22. J. Giné, and V.G. Romanovski Linearizability conditions for Lotka-Volterra planar complex cubic systems J. Phys. A 42 22 2009 225206
23. J. Giné, and V.G. Romanovski Integrability conditions for Lotka-Volterra planar complex quintic systems Nonlinear Anal. Real World Appl. 11 3 2010 2100 2105
24. J. Giné, and X. Santallusia On the Poincaré-Liapunov constants and the Poincaré series Appl. Math. (Warsaw) 28 1 2001 17 30
25. J. Giné, and X. Santallusia Implementation of a new algorithm of computation of the Poincaré-Liapunov constants J. Comput. Appl. Math. 166 2 2004 465 476
26. Ji Bin Li, and Qi Ming Huang Bifurcation of limit cycles forming compound eyes in the cubic systems Chinese Ann. Math. Ser. B 8 4 1987 391 403
27. M.A. Liapunov Problme général de la stabilité du mouvement Ann. Math. Stud. 17 1947 Pricenton University Press
28. N.G. Lloyd, and J.M. Pearson REDUCE and the bifurcation of limit cycles J. Symbolic Comput. 9 1990 215 224
29. N.G. Lloyd, and J.M. Pearson Computing center conditions for certain cubic systems J. Comput. Appl. Math. 40 1992 323 336
30. J.M. Pearson, N.G. Lloyd, and C.J. Christopher Algorithmic derivation of centre conditions SIAM Rev. 38 1996 619 636
31. H. Poincaré Mémoire sur les courbes définies par les équations différentielles J. Math. 37 1881 375 422
32. H. Poincaré Mémoire sur les courbes définies par les équations différentielles J. Math. 8 1882 251 296
33. H. Poincaré Oeuvres de Henri Poincaré vol. I 1951 Gauthier-Villars Paris pp. 3-84
34. K.S. Sibirskii On the number of limit cycles in the neighborhood of a singular point Differ. Eqn. 1 1965 36 47
35. S. Smale Mathematical problems for the next century Math. Intelligencer 20 1998 7 15
36. Shi Songling A method of constructing cycles without contact around a weak focus J. Differ. Eqn. 41 1981 301 312
37. Shi Songling On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields J. Differ. Eqn. 52 1984 52 57
38. H. oŁa̧dek On certain generalization of Bautin's theorem Nonlinearity 7 1994 273 279
39. H. oŁa̧dek Eleven small limit cycles in a cubic vector field Nonlinearity 8 1995 843 860