# Totally bounded differential polynomial systems in ${\mathbb{R}}^{2}$

Roberto Conti; Marcello Galeotti

- Volume: 13, Issue: 2, page 91-99
- ISSN: 1120-6330

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topConti, Roberto, and Galeotti, Marcello. "Totally bounded differential polynomial systems in $\mathbb{R}^{2}$." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13.2 (2002): 91-99. <http://eudml.org/doc/252440>.

@article{Conti2002,

abstract = {Totally bounded differential systems in $\mathbb\{R\}^\{2\}$ are defined as having all trajectories bounded. By Dulac’s finiteness theorem it is proved that totally bounded polynomial systems exhibit an unbounded «annulus» of cycles. The portrait of the remaining trajectories is examined in the case the system has, in $\mathbb\{R\}^\{2\}$, a unique singular point. Work is in progress concerning the study of totally bounded polynomial systems with two singular points.},

author = {Conti, Roberto, Galeotti, Marcello},

journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},

keywords = {Polynomial systems; Total boundedness; Petals; polynomial systems; total boundedness; petals},

language = {eng},

month = {6},

number = {2},

pages = {91-99},

publisher = {Accademia Nazionale dei Lincei},

title = {Totally bounded differential polynomial systems in $\mathbb\{R\}^\{2\}$},

url = {http://eudml.org/doc/252440},

volume = {13},

year = {2002},

}

TY - JOUR

AU - Conti, Roberto

AU - Galeotti, Marcello

TI - Totally bounded differential polynomial systems in $\mathbb{R}^{2}$

JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

DA - 2002/6//

PB - Accademia Nazionale dei Lincei

VL - 13

IS - 2

SP - 91

EP - 99

AB - Totally bounded differential systems in $\mathbb{R}^{2}$ are defined as having all trajectories bounded. By Dulac’s finiteness theorem it is proved that totally bounded polynomial systems exhibit an unbounded «annulus» of cycles. The portrait of the remaining trajectories is examined in the case the system has, in $\mathbb{R}^{2}$, a unique singular point. Work is in progress concerning the study of totally bounded polynomial systems with two singular points.

LA - eng

KW - Polynomial systems; Total boundedness; Petals; polynomial systems; total boundedness; petals

UR - http://eudml.org/doc/252440

ER -

## References

top- Berlinskii, A.N., On the structure of the neighborhood of a singular point of a two-dimensional autonomous system. Sov. Math., Dokl., 10, 1969, 882-885; translation from: Dokl. Akad. Nauk SSSR, 187, 1969, 502-505. Zbl0238.34049MR254320
- Ecalle, J., Finitude des cycles-limites et accéléro-sommation de l’application de retour. Proc. Meet. Luminy on Bifurcation of vector fields (Luminy, 1989). LNM 1455, Springer-Verlag, 1990, 74-159. Zbl0729.34016MR1094378DOI10.1007/BFb0085391
- Galeotti, M. - Villarini, M., Some properties of planar polynomial systems of even degree. Ann. Mat. Pura Appl., s. 4, vol. 161, 1992, 299-313. Zbl0757.34023MR1174822DOI10.1007/BF01759643
- Il'yashenko, Yu.S., Finiteness theorems for limit cycles. Russian Math. Surveys, 45, 2, 1990, 129-203; translation from: Uspekhi Mat. Nauk, 45, 2, 1990, 143-200. Zbl0731.34027MR1069351DOI10.1070/RM1990v045n02ABEH002335

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