| Title:
|
Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four (English) |
| Author:
|
Sáez, Eduardo |
| Author:
|
Stange, Eduardo |
| Author:
|
Szántó, Iván |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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57 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
105-114 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A class of degree four differential systems that have an invariant conic $ x^2+Cy^2=1$, $C\in {\mathbb{R}}$, is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems. (English) |
| Keyword:
|
stability |
| Keyword:
|
limit cycle |
| Keyword:
|
center |
| Keyword:
|
bifurcation |
| MSC:
|
34-04 |
| MSC:
|
34C05 |
| MSC:
|
58F14 |
| MSC:
|
58F21 |
| MSC:
|
92D25 |
| idZBL:
|
Zbl 1168.92319 |
| idMR:
|
MR2309952 |
| . |
| Date available:
|
2009-09-24T11:44:14Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128158 |
| . |
| Reference:
|
[1] N. N. Bautin: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus.Mat. Sbornik 30(72) (1952), 181–196. Zbl 0059.08201, MR 0045893 |
| Reference:
|
[2] I. Bendixson: Sur les courbes définies par des équations différentielles.Acta Math. 24 (1900), 1–88. (French) |
| Reference:
|
[3] W. A. Coppel: Some quadratics systems with at most one limit cycle.Dynam. report. Expositions Dynam. Systems (N.S.) 2 (1989), 61–88. MR 1000976 |
| Reference:
|
[4] D. Cozma, A. Suba: The solution of the problem of center for cubic differential systems with four invariant straight lines.An. Stiint. Univ. Al. I. Cuza Iasi, Ser. Nuova, Mat. 44 (Suppl.) (1999), 517–530. MR 1814187 |
| Reference:
|
[5] D. Cozma, A. Suba: Solution of the problem of center for a cubic differential systems with three invariant straight lines.Qual. Theory Dyn. Syst. 2 (2001), 129–143. MR 1844982, 10.1007/BF02969386 |
| Reference:
|
[6] J. Chavarriga, E. Sáez, I. Szántó, and M. Grau: Coexistence of limit cycles and invariant algebraic curves on a Kukles systems.Nonlinear Anal., Theory Methods Appl. 59 (2004), 673–693. MR 2096323, 10.1016/S0362-546X(04)00278-0 |
| Reference:
|
[7] C. Lansun, W. Mingshu: Relative position and number of limit cycles of a quadratic differential system.Acta Math. Sin. 22 (1979), 751–758. (Chinese) MR 0559742 |
| Reference:
|
[8] L. A. Cherkas, L. I. Zhilevich: The limit cycles of some differential equations.Differ. Uravn. 8 (1972), 924–929. (Russian) |
| Reference:
|
[9] C. Christopher: Quadratic systems having a parabola as an integral curve.Proc. R. Soc. Edinb. Sect. A 112 (1989), 113–134. Zbl 0677.34034, MR 1007539, 10.1017/S0308210500028195 |
| Reference:
|
[10] D. Guoren, W. Songlin: Closed orbits and straight line invariants in $E_3$ systems.Acta Mathematica Sci. 9 (1989), 251–261. (Chinese) |
| Reference:
|
[11] H. Dulac: Sur les cycles limites.S. M. F. Bull. 51 (1923), 45–188. (French) MR 1504823 |
| Reference:
|
[12] D. Hilbert: Mathematical problems.American Bull (2) 8 (1902), 437–479. MR 1557926, 10.1090/S0002-9904-1902-00923-3 |
| Reference:
|
[13] E. D. James, N. G. Lloyd: A cubic system with eigth small-amplitude limit cycles.IMA J. Appl. Math. 47 (1991), 163–171. MR 1130524, 10.1093/imamat/47.2.163 |
| Reference:
|
[14] R. Kooij: Limit cycles in polynomial systems.PhD. thesis, University of Technology, Delft, 1993. |
| Reference:
|
[15] J. Li: Hilbert’s 16th problem for $ n=3 \: H(3) \ge 11 $.Kexue Tongbao 31 (1984), 718. |
| Reference:
|
[16] A. Liénard: Etude des oscillations entreteneues.Re. générale de l’électricité 23 (1928), 901–912. (French) |
| Reference:
|
[17] Z. H. Liu, E. Sáez, and I. Szántó: A cubic systems with an invariant triangle surrounding at last one limit cycle.Taiwanese J. Math. 7 (2003), 275–281. MR 1978016, 10.11650/twjm/1500575064 |
| Reference:
|
[18] N. G. Lloyd, J. M. Pearson, E. Sáez, and I. Szántó: Limit cycles of a cubic Kolmogorov system.Appl. Math. Lett. 9 (1996), 15–18. MR 1389591, 10.1016/0893-9659(95)00095-X |
| Reference:
|
[19] N. G. Lloyd, J. M. Pearson, E. Sáez, and I. Szántó: A cubic Kolmogorov system with six limit cycles.Computers Math. Appl. 44 (2002), 445–455. MR 1912841, 10.1016/S0898-1221(02)00161-X |
| Reference:
|
[20] N. G. Lloyd, J. M. Pearson: Five limit cycles for a simple cubic system.Publications Mathematiques 41 (1997), 199–208. MR 1461651 |
| Reference:
|
[21] N. G. Lloyd, T. R. Blows, and M. C. Kalenge: Some cubic systems with several limit cycles.Nonlinearity (1988), 653–669. MR 0967475 |
| Reference:
|
[22] S. Ning, S. Ma, K. H. Kwek, and Z. Zheng: A cubic systems with eight small-amplitude limit cycles.Appl. Math. Lett. 7 (1994), 23–27. MR 1350389, 10.1016/0893-9659(94)90005-1 |
| Reference:
|
[23] H. Poincaré: Mémorie sur les courbes définies par leś équations differentialles I–VI, Oeuvre I.Gauthier-Villar, Paris, 1880–1890. (French) |
| Reference:
|
[24] L. S. Pontryagin: On dynamical systems close to Hamiltonian ones.Zh. Eksper. Teoret. Fiz. 4 (1934), 883–885. (Russian) |
| Reference:
|
[25] J. W. Reyn: A bibliography of the qualitative theory of quadratic systems of differential equation in the plane.Report TU Delft 92-17, second edition, 1992. |
| Reference:
|
[26] E. Sáez, I. Szántó, and E. González-Olivares: Cubic Kolmogorov system with four limit cycles and three invariant straight lines.Nonlinear Anal., Theory Methods Appl. 47 (2001), 4521–4525. MR 1975846, 10.1016/S0362-546X(01)00565-X |
| Reference:
|
[27] S. Songling: A concrete example of the existence of four limit cycles for planar quadratics systems.Sci. Sin. XXIII (1980), 153–158. MR 0574405 |
| Reference:
|
[28] S. Songling: System of equation ($ E_{3}$) has five limit cycles.Acta Math. Sin. 18 (1975). (Chinese) |
| Reference:
|
[29] S. Guangjian, S. Jifang: The $n$-degree differential system with $\frac{1}{2}{(n-1)(n+1)}$ straight line solutions has no limit cycles.Proc. Conf. Ordinary Differential Equations and Control Theory, Wuhan 1987 (1987), 216–220. (Chinese) MR 1043472 |
| Reference:
|
[30] B. van der Pol: On relaxation-oscillations.Philos. Magazine 7 (1926), 978–992. 10.1080/14786442608564127 |
| Reference:
|
[31] W. Dongming: A class of cubic differential systems with 6-tuple focus.J. Differ. Equations 87 (1990), 305–315. Zbl 0712.34044, MR 1072903, 10.1016/0022-0396(90)90004-9 |
| Reference:
|
[32] Y. Xinan: A survey of cubic systems.Ann. Differ. Equations 7 (1991), 323–363. Zbl 0747.34019, MR 1139341 |
| Reference:
|
[33] Y. Yanqian, Y. Weiyin: Cubic Kolmogorov differential system with two limit cycles surrounding the same focus.Ann. Differ. Equations 1 (1985), 201–207. MR 0834242 |
| Reference:
|
[34] P. Yu and M. Han: Twelve limit cycles in a cubic order planar system with $Z_2$-symmetry.Communications on pure and applied analysis 3 (2004), 515–526. MR 2098300 |
| Reference:
|
[35] H. Zoladek: Eleven small limit cycles in a cubic vector field.Nonlinearity 8 (1995), 843–860. Zbl 0837.34042, MR 1355046, 10.1088/0951-7715/8/5/011 |
| Reference:
|
[36] W. Stephen: A System for Doing Mathematics by Computer.Wolfram Research Mathematica, 1988. Zbl 0671.65002 |
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