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Bifurcations of limit cycles from infinity for a class of quintic polynomial system. (English) Zbl 1070.34064

The paper is devoted to a class of quintic systems. By means of special transformations infinity is mapped turn into the origin. Having found the relation between focal values and singular point values, the authors investigate bifurcations of limit cycles of the original system from infinity. In such way, a quintic system with a small parameter and eight other parameters can be constructed, where 8 limit cycles can bifurcate from infinity.

MSC:

34C23 Bifurcation theory for ordinary differential equations
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
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References:

[1] Li, J., Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifurcation and Chaos, 13, 47-106 (2003) · Zbl 1063.34026
[2] Blows, T. R.; Rousseau, C., Bifurcation at infinity in polynomial vector fields, J. Differential Equations, 104, 215-242 (1993) · Zbl 0778.34024
[3] Liu, Y., Theory of center-focus for a class of higher-degree critical points and infinite points, Science in China (Series A), 44, 37-48 (2001)
[4] Liu, Y.; Chen, H., Stability and bifurcations of limit cycles of the equator in a class of cubic polynomial systems, Comput. Math. Appl., 44, 997-1005 (2002) · Zbl 1084.34523
[5] Cheng, H.; Liu, Y., Limit cycle of the equator in a quintic polynomial system, Chinese Ann. Math., 24A, 219-224 (2003), (in Chinese) · Zbl 1048.34063
[6] Liu, Y.; Li, J., Theory of values of singular point in complex autonomous differential system, Science in China (Series A), 33, 10-24 (1990) · Zbl 0686.34027
[7] Liu, Y.; Chen, H., Formulas of singular point quantities and the first 10 saddle quantities for a class of cubic system, Acta Math. Appl. Sinica, 25, 295-302 (2002), (in Chinese) · Zbl 1014.34021
[8] Gobber, F.; Willamowski, K., Liapunov approach to multiple Hopf bifurcation, J. Math. Anal. Appl., 71, 330-350 (1979) · Zbl 0444.34040
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