Seven large-amplitude limit cycles in a cubic polynomial system. (English) Zbl 1111.37035

Summary: The problem of limit cycles bifurcated from the equator for a cubic polynomial system is investigated. The best result so far in the literature for this problem is six limit cycles. By using the method of singular point value, we prove that a cubic polynomial system can bifurcate seven limit cycles from the equator. We also find that a rational system has an isochronous center at the equator.


37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37C10 Dynamics induced by flows and semiflows
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[1] Bautin N. N., Amer. Math. Soc. Trans. 100 pp 397–
[2] DOI: 10.1006/jdeq.1993.1070 · Zbl 0778.34024
[3] Cheng H. B., Chinese Ann. Math. A 24 pp 219–
[4] Gobber F., J. Math. Anal. Appl. 71 pp 330–
[5] DOI: 10.1016/j.bulsci.2004.02.002 · Zbl 1070.34064
[6] James E. M., I.M.A.J. Applied Math. 47 pp 163–
[7] Liu Y. R., Science in China (Series A) 44 pp 37–
[8] DOI: 10.1016/S0898-1221(02)00209-2 · Zbl 1084.34523
[9] Liu Y. R., Acta Math. Appl. Sin. 25 pp 295–
[10] DOI: 10.1016/S0007-4497(02)00006-4 · Zbl 1034.34032
[11] Liu Y. R., Science in China (Series A) 33 pp 10–
[12] DOI: 10.1016/0893-9659(94)90005-1 · Zbl 0804.34033
[13] DOI: 10.1088/0951-7715/8/5/011 · Zbl 0837.34042
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