An improvement of the non-existence region for limit cycles of the Bogdanov-Takens system

Makoto Hayashi

Archivum Mathematicum (2020)

  • Volume: 056, Issue: 4, page 199-206
  • ISSN: 0044-8753

Abstract

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In this paper, an improvement of the global region for the non-existence of limit cycles of the Bogdanov-Takens system, which is well-known in the Bifurcation Theory, is given by two ideas. The first is to apply the existence of the algebraic invariant curve of the system to the Bendixson-Dulac criterion, and the second is to consider a necessary condition in order that a closed orbit of the system includes two equilibrium points. In virtue of these methods, it shall be shown that our previous result and the result of Gasull et al. are improved partially.

How to cite

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Hayashi, Makoto. "An improvement of the non-existence region for limit cycles of the Bogdanov-Takens system." Archivum Mathematicum 056.4 (2020): 199-206. <http://eudml.org/doc/297240>.

@article{Hayashi2020,
abstract = {In this paper, an improvement of the global region for the non-existence of limit cycles of the Bogdanov-Takens system, which is well-known in the Bifurcation Theory, is given by two ideas. The first is to apply the existence of the algebraic invariant curve of the system to the Bendixson-Dulac criterion, and the second is to consider a necessary condition in order that a closed orbit of the system includes two equilibrium points. In virtue of these methods, it shall be shown that our previous result and the result of Gasull et al. are improved partially.},
author = {Hayashi, Makoto},
journal = {Archivum Mathematicum},
keywords = {Liénard system; Bogdanov-Takens system; limit cycle; Bendixson-Dulac criterion; algebraic invariant curve},
language = {eng},
number = {4},
pages = {199-206},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {An improvement of the non-existence region for limit cycles of the Bogdanov-Takens system},
url = {http://eudml.org/doc/297240},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Hayashi, Makoto
TI - An improvement of the non-existence region for limit cycles of the Bogdanov-Takens system
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 4
SP - 199
EP - 206
AB - In this paper, an improvement of the global region for the non-existence of limit cycles of the Bogdanov-Takens system, which is well-known in the Bifurcation Theory, is given by two ideas. The first is to apply the existence of the algebraic invariant curve of the system to the Bendixson-Dulac criterion, and the second is to consider a necessary condition in order that a closed orbit of the system includes two equilibrium points. In virtue of these methods, it shall be shown that our previous result and the result of Gasull et al. are improved partially.
LA - eng
KW - Liénard system; Bogdanov-Takens system; limit cycle; Bendixson-Dulac criterion; algebraic invariant curve
UR - http://eudml.org/doc/297240
ER -

References

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  2. Gasull, A., Giacomini, H., Torregrosa, J., 10.1088/0951-7715/23/12/001, Nonlinearity 23 (2010), 2977–3001. (2010) MR2734501DOI10.1088/0951-7715/23/12/001
  3. Gasull, A., Guillamon, A., Non-existence of limit cycles for some predator–prey systems, Proceedings of Equadiff'91, World Scientific, Singapore, 1993, pp. 538–546. (1993) MR1242294
  4. Hayashi, M., 10.1007/s10012-000-0225-0, Southeast Asian Bull. Math. 24 (2000), 225–229. (2000) MR1810059DOI10.1007/s10012-000-0225-0
  5. Hayashi, M., A global condition for the non-existence of limit cycles of Bogdanov-Takens system, Far East J. Math. Sci. 14 (1) (2004), 127–136. (2004) MR2096965
  6. Hayashi, M., Villari, G., Zanolin, F., 10.14232/ejqtde.2018.1.55, Electron. J. Qual. Theory Differ. Equ. 55 (2018), 1–10. (2018) MR3827993DOI10.14232/ejqtde.2018.1.55
  7. Kuznetsov, Y., Elements of Applied Bifurcation Theory, second ed., Springer-Verlag, New York, 1998. (1998) Zbl0914.58025MR1711790
  8. Li, Cheng-zhi, Rousseau, C., Wang, X., 10.4153/CMB-1990-015-3, Canad. Math. Bull. 33 (1) (1990), 84–92. (1990) MR1036862DOI10.4153/CMB-1990-015-3
  9. Matsumoto, T., Komuro, M., Kokubu, H., Tokunaga, R., Bifurcations (Sights, Sounds and Mathematics), Springer-Verlag, New York, 1993. (1993) 
  10. Perko, L.M., 10.1137/0152069, SIAM J. Appl. Math. 52 (1992), 1172–1192. (1992) MR1174053DOI10.1137/0152069
  11. Perko, L.M., 10.1007/978-1-4613-0003-8, Texts Appl. Math., vol. 7, Springer-Verlag, New York, 3rd ed., 2001. (2001) MR1801796DOI10.1007/978-1-4613-0003-8
  12. Roussarie, R., Wagener, F., A study of the Bogdanov-Takens bifurcation, Resenhas 2 (1995), 1–25. (1995) MR1358328
  13. Teschl, G., 10.1090/gsm/140/06, Graduate Studies in Mathematics, vol. 140, AMS, Providence, 2012. (2012) MR2961944DOI10.1090/gsm/140/06
  14. Zhi-fen, Z., Tong-ren, D., Wen-zao, H., Zhen-xi, D., Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, vol. 102, AMS, Providence, 1992. (1992) MR1175631

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