×

A cubic system with a limit cycle bounded by two invariant parabolas. (English) Zbl 1207.34038

Summary: We show the existence of a cubic system having at least one limit cycle bounded by two invariant parabolas. We also obtain necessary and sufficient conditions for the critical point in the interior of the bounded region to be a center.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations

Software:

Mathematica; Matlab
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Coppel W. A., Some quadratic systems with at most one limit cycle, Dynam. report. Expositions Dynam. Systems (N.S.) 2, 61–88, (1989)
[2] Cozma D. and Suba A., The solution of the problem of center for cubic differential systems with four invariant straight lines, Analele Stiintifice, Tom XLIV, Suppl., 517–530, (1998) · Zbl 1009.34026
[3] Cozma D, and Suba A., Solution of the problem of the center for a cubic differential system with three invariant straight lines, Qualitative Theory of Dynamical Systems, 2(1), 129–143, (2001) · Zbl 0994.34021
[4] Chavarriga J., Sáez E., Szántó I. and Grau M., Coexistence of limit cycles and invariant algebraic curves on a Kukles systems, Nonlinear Analysis, Theory, Methods and Applications, 67(4), 1005–1014, (2007) · Zbl 1124.34017
[5] Cherkas L. A. and Zhilevich L. I., The limit cycles of certain differential equations, Differentsial’nye Urawneniya 8 (Russian), 1207–1213, (1972)
[6] Christopher C., Quadratic systems having a parabola as an integral curve, Proc. Roy. Soc. Edinburgh Sect. A, 112, 113–134, (1989) · Zbl 0677.34034
[7] Guoren D. and Songlin W., Closed orbits and straight line invariants in E 3 systems, Acta Mathematica Scientia, (Chinese) 9, 251–261, (1989)
[8] Hilbert D., Mathematische Problems (lecture), Second Internat. Congress Math Paris 1900, Nachr. Ges. Wiss. Gttingen Math.Phys. Kl., 253–297, (1900) · JFM 31.0068.03
[9] Kooij R., Limit cycles in polynomial systems, thesis, University of Technology, Delft (1993) · Zbl 0938.34513
[10] Liu Z. H., Sáez E. and Szántó I., A cubic systems with an invariant triangle surrounding at least one limit cycle, Taiwanese Journal of Mathematics, 7(2), 275–281, (2003) · Zbl 1051.34025
[11] Lloyd N. G., Pearson J. M., Sáez E. and Szántó I., Limit cycles of a cubic Kolmogorov system, Applied Mathematics Letters, 9, 15–18, (1996) · Zbl 0858.34023
[12] Lloyd N. G., Pearson J. M., Sáez E. and Szántó I., A cubic Kolmogorov system with six limit cycles, Computers and Mathematics with applications, 44(3–4), 445–455, (2002) · Zbl 1210.34048
[13] MATLAB: The Language of technical computing, Using MATLAB (version 7.0), Mat-Works, Natwick, MA (2004)
[14] Sáez E., Szántó I. and González-Olivares E., Cubic Kolmogorov system with four limit cycles and three invariant straight lines, Nonlinear Analysis, 47(7), 4521–4525, (2001) · Zbl 1042.34547
[15] Guangjian S. and Jifang S, The n-degree differential system with (n)(n+1)/2 straight line solutions has no limit cycles, Proc. of Ordinary Differential Equations and Control Theory, Wuhan, (Chinese) 216–220, (1987)
[16] Xinan Y., A survey of cubic systems, Ann. Differential Equations 7(3), 323–363, (1991) · Zbl 0747.34019
[17] Yanqian Y. and Weiyin Y. Cubic Kolmogorov Differential system with two limit cycles surrounding the same focus, Ann. of Diff., Eqs., 1(2), 201–207, (1985) · Zbl 0597.34020
[18] Ye Yanqian Cubic Kolmogorov system having three straight line integrals forming a triangle, Ann. of Diff. Eqs., 6(3), 365–372, (1990) · Zbl 0728.34029
[19] Wolfram Research Mathematica, A System for Doing Mathematics by Computer, Champaign, IL (1988)
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.