PapersNo Access

MONODROMY AND STABILITY FOR NILPOTENT CRITICAL POINTS

M. J. ÁLVAREZ

Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain

Author for correspondence.

Partially supported by DGES No. BFM2002-04236-C02-2 and CONSCIT 2001SGR-00173.

Search for more papers by this author

and

A. GASULL

Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Partially supported by DGES No. BFM2002-04236-C02-2 and CONSCIT 2001SGR-00173.

Search for more papers by this author

https://doi.org/10.1142/S0218127405012740Cited by:71 (Source: Crossref)

Abstract

We give a new and short proof of the characterization of monodromic nilpotent critical points. We also calculate the first generalized Lyapunov constants in order to solve the stability problem. We apply the results to several families of planar systems obtaining necessary and sufficient conditions for having a center. Our method also allows us to generate limit cycles from the origin.

Keywords:

References

Álvarez, M. J. & Gasull, A. [2005] "Generating limit cycles from a nilpotent critical point by using its normal form," preprint . Google Scholar
M. J. M. Alwash and N. G. Lloyd, Proc. Roy. Soc. Edinburgh A 105, 129 (1987). Crossref, Web of Science, Google Scholar
A. F. Andreev, Akad. Nauk SSSR. Prikl. Mat. Meh. 17, 333 (1953). Google Scholar
A. F. Andreev, AMS Transl. 8, 183 (1958). Google Scholar
A. F. Andreev, A. P. Sadovskii and V. A. Tsikalyuk, Diff. Eqs. 39, 155 (2003). Crossref, Web of Science, Google Scholar
E. Batcheva and N. B. Medvedeva, Electron. J. Qualit. Th. Diff. Eqs. 19, 1 (2000). Web of Science, Google Scholar
J. Chavarrigaet al., Ergod. Th. Dyn. Syst. 23, 417 (2003). Web of Science, Google Scholar
A. Gasull, A. Guillamon and V. Mañosa, J. Math. Anal. Appl. 211, 190 (1997). Crossref, Web of Science, Google Scholar
A. Gasull and J. Torregrosa, Nonlin. Anal. 47, 4479 (2001). Crossref, Web of Science, Google Scholar
A. Gasull, V. Mañosa and F. Mañosas, J. Diff. Eqs. 182, 169 (2002). Crossref, Web of Science, Google Scholar
A. M. Lyapunov , Stability of Motion, Mathematics in Science and Engineering 30 ( Academic Press , NY–London , 1966 ) . Google Scholar
V. Mañosa, Int. J. Bifurcation and Chaos 12, 687 (2002). Link, Web of Science, Google Scholar
N. B. Medvedeva, Sib. Math. J. 33, 280 (1992). Crossref, Web of Science, Google Scholar
R. Moussu, Ergod. Th. Dyn. Syst. 2, 241 (1982). Crossref, Google Scholar
A. P. Sadovskii, Diff. Eqs. 4, 2002 (1968). Google Scholar
A. P. Sadovskii, Diff. Eqs. 7, 2285 (1971). Google Scholar