×

Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four. (English) Zbl 1168.92319

Summary: A class of degree four differential systems with 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.

MSC:

92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models

Software:

Mathematica
PDF BibTeX XML Cite
Full Text: DOI EuDML Link

References:

[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; English translation in AMS 1(5) (1962), 396–413. · Zbl 0059.08201
[2] I. Bendixson: Sur les courbes définies par des équations différentielles. Acta Math. 24 (1900), 1–88. (In French.) · JFM 31.0328.03
[3] W. A. Coppel: Some quadratics systems with at most one limit cycle. Dynam. report. Expositions Dynam. Systems (N.S.) 2 (1989), 61–88.
[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.
[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. · Zbl 0994.34021
[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. · Zbl 1076.34029
[7] C. Lansun, W. Mingshu: Relative position and number of limit cycles of a quadratic differential system. Acta Math. Sin. 22 (1979), 751–758. (In Chinese.) · Zbl 0433.34022
[8] L. A. Cherkas, L. I. Zhilevich: The limit cycles of some differential equations. Differ. Uravn. 8 (1972), 924–929. (In Russian.) · Zbl 0289.34035
[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
[10] D. Guoren, W. Songlin: Closed orbits and straight line invariants in E 3 systems. Acta Mathematica Sci. 9 (1989), 251–261. (In Chinese.)
[11] H. Dulac: Sur les cycles limites. S. M. F. Bull. 51 (1923), 45–188. (In French.) · JFM 49.0304.01
[12] D. Hilbert: Mathematical problems. American Bull (2) 8 (1902), 437–479. · JFM 33.0976.07
[13] E. D. James, N. G. Lloyd: A cubic system with eigth small-amplitude limit cycles. IMA J. Appl. Math. 47 (1991), 163–171. · Zbl 0743.34037
[14] R. Kooij: Limit cycles in polynomial systems. PhD. thesis. University of Technology, Delft, 1993. · Zbl 0938.34513
[15] J. Li: Hilbert’s 16th problem for n = 3: H(3) 11. Kexue Tongbao 31 (1984), 718.
[16] A. Liénard: Etude des oscillations entreteneues. Re. générale de l’électricité 23 (1928), 901–912. (In French.)
[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. · Zbl 1051.34025
[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. · Zbl 0858.34023
[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. · Zbl 1210.34048
[20] N. G. Lloyd, J. M. Pearson: Five limit cycles for a simple cubic system. Publications Mathematiques 41 (1997), 199–208. · Zbl 0885.34029
[21] N. G. Lloyd, T. R. Blows, and M. C. Kalenge: Some cubic systems with several limit cycles. Nonlinearity (1988), 653–669. · Zbl 0691.34024
[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. · Zbl 0804.34033
[23] H. Poincaré: Mémorie sur les courbes définies par lés équations differentialles I–VI, Oeuvre I. Gauthier-Villar, Paris, 1880–1890. (In French.)
[24] L. S. Pontryagin: On dynamical systems close to Hamiltonian ones. Zh. Eksper. Teoret. Fiz. 4 (1934), 883–885. (In Russian.)
[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.
[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. · Zbl 1042.34547
[27] S. Songling: A concrete example of the existence of four limit cycles for planar quadratics systems. Sci. Sin. XXIII (1980), 153–158. · Zbl 0431.34024
[28] S. Songling: System of equation (E 3) has five limit cycles. Acta Math. Sin. 18 (1975). (In Chinese.)
[29] S. Guangjian, S. Jifang: The n-degree differential system with 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. (In Chinese.)
[30] B. van der Pol: On relaxation-oscillations. Philos. Magazine 7 (1926), 978–992. · JFM 52.0450.05
[31] W. Dongming: A class of cubic differential systems with 6-tuple focus. J. Differ. Equations 87 (1990), 305–315. · Zbl 0712.34044
[32] Y. Xinan: A survey of cubic systems. Ann. Differ. Equations 7 (1991), 323–363. · Zbl 0747.34019
[33] Y. Yanqian, Y. Weiyin: Cubic Kolmogorov differential system with two limit cycles surrounding the same focus. Ann. Differ. Equations 1 (1985), 201–207. · Zbl 0597.34020
[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. · Zbl 1085.34028
[35] H. Zoladek: Eleven small limit cycles in a cubic vector field. Nonlinearity 8 (1995), 843–860. · Zbl 0837.34042
[36] W. Stephen: A System for Doing Mathematics by Computer. Wolfram Research Mathematica, 1988. · Zbl 0671.65002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.