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Liénard system; Bogdanov-Takens system; limit cycle; Bendixson-Dulac criterion; algebraic invariant curve
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.
[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. DOI 10.1016/j.jde.2013.07.006 | MR 3090073
[2] Gasull, A., Giacomini, H., Torregrosa, J.: Some results on homoclinic and heteroclinic connections in planar systems. Nonlinearity 23 (2010), 2977–3001. DOI 10.1088/0951-7715/23/12/001 | MR 2734501
[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
[4] Hayashi, M.: On the Non-existence of the closed Orbit for a Liénard system. Southeast Asian Bull. Math. 24 (2000), 225–229. DOI 10.1007/s10012-000-0225-0 | MR 1810059
[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
[6] Hayashi, M., Villari, G., Zanolin, F.: On the uniqueness of limit cycle for certain Liénard systems without symmetry. Electron. J. Qual. Theory Differ. Equ. 55 (2018), 1–10. DOI 10.14232/ejqtde.2018.1.55 | MR 3827993
[7] Kuznetsov, Y.: Elements of Applied Bifurcation Theory. second ed., Springer-Verlag, New York, 1998. MR 1711790 | Zbl 0914.58025
[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. DOI 10.4153/CMB-1990-015-3 | MR 1036862
[9] Matsumoto, T., Komuro, M., Kokubu, H., Tokunaga, R.: Bifurcations (Sights, Sounds and Mathematics). Springer-Verlag, New York, 1993.
[10] Perko, L.M.: A global analysis of the Bogdanov-Takens system. SIAM J. Appl. Math. 52 (1992), 1172–1192. DOI 10.1137/0152069 | MR 1174053
[11] Perko, L.M.: Differential Equations and Dynamical Systems. Texts Appl. Math., vol. 7, Springer-Verlag, New York, 3rd ed., 2001. DOI 10.1007/978-1-4613-0003-8 | MR 1801796
[12] Roussarie, R., Wagener, F.: A study of the Bogdanov-Takens bifurcation. Resenhas 2 (1995), 1–25. MR 1358328
[13] Teschl, G.: Ordinary Differential Equations and Dynamical systems. Graduate Studies in Mathematics, vol. 140, AMS, Providence, 2012. DOI 10.1090/gsm/140/06 | MR 2961944
[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. MR 1175631
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