×

New study on the center problem and bifurcations of limit cycles for the Lyapunov system. II. (English) Zbl 1179.34030

Summary: The center problem and bifurcations of limit cycles for the Lyapunov system are continuously studied. We shall prove that we can construct successively a formal series such that the Lyapunov system is reduced a half-normal form. From the coefficients of the half-normal form, we obtain directly the Lyapunov constants of the origin. As examples, for two classes of cubic systems, the center and focus problem, and multiple bifurcations of limit cycles are studied.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C23 Bifurcation theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1142/S0218127405012740 · Zbl 1088.34021 · doi:10.1142/S0218127405012740
[2] DOI: 10.1016/j.jmaa.2005.05.064 · Zbl 1100.34030 · doi:10.1016/j.jmaa.2005.05.064
[3] Liu Y. R., China 33 pp 10– (1989)
[4] Liu Y. R., China 31 pp 37– (2001)
[5] Liu Y. R., Int. J. Bifurcation and Chaos 19. (2009)
[6] DOI: 10.1017/S0143385700001553 · Zbl 0509.34027 · doi:10.1017/S0143385700001553
[7] DOI: 10.1006/jdeq.2001.4043 · Zbl 1005.34034 · doi:10.1006/jdeq.2001.4043
[8] DOI: 10.1007/BF02684366 · Zbl 0279.58009 · doi:10.1007/BF02684366
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.