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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: 1572-9141 (online)
Volume: 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:
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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