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A polynomial differential system with nine limit cycles at infinity.(English)Zbl 1154.34322

Summary: We give a recursive formula to compute the singular point quantities of a class of seventh-order polynomial systems. The first eleven singular point quantities have been computed with computer algebra systemMathematica, and the conditions for infinity to be a center have been deduced as well. At last, we construct a system that allows the appearance of nine limit cycles in the neighborhood of infinity.

MSC:

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
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems

Software:

Mathematica

Cite

References:

[1]Zolsdek, H., Eleven small limit cycles in a cubic vector field, Nonlinearity, 8, 843-860 (1995) ·Zbl 0837.34042
[2]James, E. M.; Lloyd, N. G., A cubic system with eight small-amplitude limit cycles, I.M.A.J. Applied Math., 47, 163-171 (1991) ·Zbl 0743.34037
[3]Ye, Y., Qualitative Theory of Polynomial Differential Systems (1995), Shanghai Sci. Tech. Publ.: Shanghai Sci. Tech. Publ. Philadelphia, PA, (in Chinese) ·Zbl 0854.34003
[4]Blows, T. R.; Rousseau, C., Bifurcation at infinity in polynomial vector fields, Journal of Differential Equations, 104, 215-242 (1993) ·Zbl 0778.34024
[5]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)
[6]Liu, Y.; Meichun, Z., Stability and bifurcation of limit cycles of the equator in a class of fifth polynomial systems, Chinese Journal of Contemporary Mathematics, 23, 1 (2002)
[7]Liu, Y.; Chen, H., Stability and bifurcations of limit cycles of the equator in a class of cubic polynomial systems, Computers Math. Applic., 44, 8, 997-1005 (2002) ·Zbl 1084.34523
[8]Liu, Y.; Li, J., Theory of values of singular point in complex autonomous differential system, Science in China (Series A), 33, 1, 10-24 (1990) ·Zbl 0686.34027
[9]Lloyd, N. G.; Pearson, J. M., Symmetry in planar dynamical systems, J. Symbolic Computation, 33, 357-366 (2002) ·Zbl 1003.34026
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.
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