| Title:
|
An improvement of the non-existence region for limit cycles of the Bogdanov-Takens system (English) |
| Author:
|
Hayashi, Makoto |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
56 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
199-206 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Liénard system |
| Keyword:
|
Bogdanov-Takens system |
| Keyword:
|
limit cycle |
| Keyword:
|
Bendixson-Dulac criterion |
| Keyword:
|
algebraic invariant curve |
| MSC:
|
34C07 |
| MSC:
|
34C25 |
| MSC:
|
34C26 |
| MSC:
|
34D20 |
| idZBL:
|
Zbl 07285960 |
| idMR:
|
MR4173074 |
| DOI:
|
10.5817/AM2020-4-199 |
| . |
| Date available:
|
2020-10-30T16:37:23Z |
| Last updated:
|
2021-02-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148387 |
| . |
| Reference:
|
[1] Gasull, A., Giacomini, H., Pérez-González, S., Torregrosa, J.: A proof of Perko’s conjectures for the Bogdanov-Takens system.J. Differential Equations 255 (2013), 2655–2671. MR 3090073, 10.1016/j.jde.2013.07.006 |
| Reference:
|
[2] Gasull, A., Giacomini, H., Torregrosa, J.: Some results on homoclinic and heteroclinic connections in planar systems.Nonlinearity 23 (2010), 2977–3001. MR 2734501, 10.1088/0951-7715/23/12/001 |
| Reference:
|
[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. MR 1242294 |
| Reference:
|
[4] Hayashi, M.: On the Non-existence of the closed Orbit for a Liénard system.Southeast Asian Bull. Math. 24 (2000), 225–229. MR 1810059, 10.1007/s10012-000-0225-0 |
| Reference:
|
[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. MR 2096965 |
| Reference:
|
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| Reference:
|
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| Reference:
|
[8] Li, Cheng-zhi, Rousseau, C., Wang, X.: A simple proof for the unicity of the limit cycle in the Bogdanov-Takens system.Canad. Math. Bull. 33 (1) (1990), 84–92. MR 1036862, 10.4153/CMB-1990-015-3 |
| Reference:
|
[9] Matsumoto, T., Komuro, M., Kokubu, H., Tokunaga, R.: Bifurcations (Sights, Sounds and Mathematics).Springer-Verlag, New York, 1993. |
| Reference:
|
[10] Perko, L.M.: A global analysis of the Bogdanov-Takens system.SIAM J. Appl. Math. 52 (1992), 1172–1192. MR 1174053, 10.1137/0152069 |
| Reference:
|
[11] Perko, L.M.: Differential Equations and Dynamical Systems.Texts Appl. Math., vol. 7, Springer-Verlag, New York, 3rd ed., 2001. MR 1801796, 10.1007/978-1-4613-0003-8 |
| Reference:
|
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| Reference:
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| Reference:
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