A cubic Kolmogorov system with six limit cycles. (English) Zbl 1210.34048

The paper is devoted to a class of cubic Kolmogorov systems, where at most six small amplitude limit cycles can bifurcate from singular point in the first quadrant. The Reduce and Maple computer systems were used for the calculation of the focal values. There are also some conclusions about the number of invariant lines.


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


Full Text: DOI


[1] Hilbert, D., Mathematical problems, Bull. amer. math. soc., 8, 437-479, (1902) · JFM 33.0976.07
[2] ()
[3] Smale, S., Mathematical problems for the next century, Math. intelligencer, 20, 2, 7-15, (1998) · Zbl 0947.01011
[4] Christopher, C., Quadratic systems having a parabola as an integral curve, Proc. roy. soc. Edinburgh, 112A, 113-134, (1989) · Zbl 0677.34034
[5] Coppel, W.A., Some quadratic systems with at most one limit cycle, Dynam. report. expositions dynam. systems (N.S.), 2, 61-88, (1989) · Zbl 0674.34026
[6] Bautin, N.N., On periodic solutions of systems of differential equations (in Russian), Prikl. mat. mekh., 18, 128, (1954)
[7] Cherkas, L.A.; Zhilevich, L.I., The limit cycles of certain differential equations (in Russian), Differ. uravn., 8, 1207-1213, (1972)
[8] Ye, Y.; Ye, W., Cubic Kolmogorov systems with two limit cycles surrounding the same focus, Ann. differential equations, 1, 2, 201-207, (1985) · Zbl 0597.34020
[9] Suo, G.; Sun, J., The n-degree differential systems with (n−1)(n+2)/2 straight line solutions have no limit cycles (in Chinese), (), 216-220
[10] Kooij, R.E., Limit cycles in polynomial systems, () · Zbl 0814.34024
[11] Lloyd, N.G.; Pearson, J.M.; Saez, E.; Szanto, I., Limit cycles of a cubic Kolmogorov system, Appl. math. lett., 9, 1, 15-18, (1996) · Zbl 0858.34023
[12] Sokulski, J., On the number of invariant lines for polynomial vector fields, Nonlinearity, 9, 479-486, (1996) · Zbl 0888.34019
[13] Zhang, X., Number of integral lines of polynomial systems of degree three and four, J. nanying university, math. biquarterly, 10, 204-212, (1993)
[14] Artés, J.C.; Grünbaum, B.; Llibre, J., On the number of invariant straight lines for polynomial differential systems, Pacific J. math., 184, 207-230, (1998) · Zbl 0930.34022
[15] Dai, G.; Wo, S., Closed orbits and straight line invariants in E3 systems (in Chinese), Acta math. sci., 9, 251-261, (1989)
[16] Wang, D., Polynomial systems for certain differential equations, J. symbolic comput., 28, 303-315, (1999) · Zbl 0948.34018
[17] Pearson, J.M.; Lloyd, N.G.; Christopher, C.J., Algorithmic derivation of center conditions, SIAM rev., 619-636, (1996) · Zbl 0876.34033
[18] Lloyd, N.G.; Pearson, J.M., REDUCE and the bifurcation of limit cycles, J. symbolic comput., 9, 215-224, (1990) · Zbl 0702.68072
[19] Collins, G.E., Subresultants and reduced polynomial remainder sequences, J. ACM, 14, 128-142, (1967) · Zbl 0152.35403
[20] C.J. Christopher, Private communication.
[21] Albarakati, W.A., Polynomial differential systems: limit cycles, centres and invariant curves, ()
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.