×

Stability and bifurcations of limit cycles of the equator in a class of cubic polynomial systems. (English) Zbl 1084.34523

After a short summary of the theory of real planar polynomial systems of autonomous differential equations, the authors consider a special class of cubic systems with an equilibrium of focal type at infinity. The recursive calculation of focal values (partly by MATHEMATICA) allows to create an example with 6 limit cycles in a neighborhood of infinity.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34D20 Stability of solutions to ordinary differential equations

Software:

Mathematica
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Shi, S., A concrete example of the existence of four limit cycles for quadratic systems, Scientia Sinica (Engl.Ed), 23, 154-158 (1980) · Zbl 0431.34024
[2] Rousseau, C., Bifurcation methods in quadratic system, (Proceedings of the Conference Oscillations, Bifurcations and Chaos of the Canadian Math. Soc., Vol. 8. Proceedings of the Conference Oscillations, Bifurcations and Chaos of the Canadian Math. Soc., Vol. 8, Toronto, 1986 (1987), CMS-AMS: CMS-AMS Providence), 637-653 · Zbl 0649.34036
[3] Blows, T. R.; Rousseau, C., Bifurcation at infinity in polynomial vector fields, J.D.E., 104, 215-242 (1993) · Zbl 0778.34024
[4] Han, M., Stability of the equator and bifurcation of limit cycles, World Scientific (1993) · Zbl 0943.34502
[5] Sotomayor, J.; Paterlini, R., Bifurcations of polynomial vector fields in the plane, (Proceedings of the Conference Oscillations, Bifurcations and Chaos of the Canadian Math. Soc., Vol. 8. Proceedings of the Conference Oscillations, Bifurcations and Chaos of the Canadian Math. Soc., Vol. 8, Toronto, 1986 (1987), CMS-AMS: CMS-AMS Providence), 665-685 · Zbl 0632.34056
[6] Liu, Y., Sci. China, Ser. A, 31, 1, 37-48 (2001)
[7] Liu, Y.; Li, J., Theory of values of singular point in complex autonomous differential systems, Sci. China. Ser. A, 19, 3, 245-255 (1989)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.