Cubic Darboux systems with a non-elementary singular point at the Poincar´ e equator
https://doi.org/10.25587/SVFU.2023.68.43.004
Abstract
We study the global behavior of the trajectories of the polynomial system˙
x = x −x2y + pxy2 + y3, ˙
y = y + py3, p ∈R.
Our study is related to the paper arXiv:2106/07516v2 [math.DS].
About the Author
E. P. Volokitin
Sobolev Institute of Mathematics
Russian Federation
Russian Federation
Evgenii P. Volokitin
4 Koptyug Avenue, 630090 Novosibirsk
References
1. Bendjeddou A., Llibre J., and Salhi T., “Dynamics of the polynomial differential systems with homogeneous nonlinearities and a star node,” J. Differ. Equ., 254, No. 8, 3530–3537 (2013).
2. Berlinskii A. N., “The qualitative study of the differential equation dy/dx = (y + a0x2 + a1xy + a2y2)/(x + b0x2 + b1xy + b2y2) [in Russian],” Differ. Uravn., 2, 353–360 (1966).
3. Yan Qian Ye, et al., Theory of Limit Cycles, Amer. Math. Soc., Providence, RI (1986) (Transl. Math. Monogr.; vol. 66).
4. Volokitin E. P. and Cheresiz V. M., “The qualitative analysis of the plane polynomial Darboux systems [in Russian],” Sib. Electron. Math. Rep., 13, 1170–1186 (2016).
5. Volokitin E. P. and Cheresiz V. M., “Dynamics of the cubic Darboux systems [in Russian],” Sib. Electron. Math. Rep., 14, 889–902 (2017).
6. Alarcon B., Castro S. B. S. D., and Labouriau I. S., “Global planar dynamics with star nodes: beyond Hilbert’s 16th problem,” arXiv:2106/07516v2 [math.DS]
7. Volokitin E. P., “Singular points od polynomial Darboux systems,” Qual. Theory Dyn. Syst., 18, No. 3, 909–930 (2019).
8. Andronov A. A., et al., The Qualitative Theory of Dynamical Systems in a Plane [in Russian], Nauka, Moscow (1956).
9. Dumortier F., et al., Qualitative Theory of Planar Differential Systems, Springer, Berlin; Heidelberg; New York (2006).
10. Arnol’d V. I., Additional Chapters on the Theory of Ordinary Differential Equations [in Russian], Nauka, Mocsow (1978).
11. Berezovskaya F. S. and Kreutzer G. P., Complex Singular Points of the System of Two Differential Equations [in Russian], ONTI NTsBI AN SSSR, Pushchino (1985).
12. Garc´ıa A. and Grau M., “A survey on the inverse integrating factor,” Qual. Theory Dyn. Syst., 9, 115–166 (2010).
13. Euler L., Introduction to Infinitesimal Analysis [in Russian], vol. II, Fizmatgiz, Moscow (1961).
Review
For citations:
Volokitin E.P. Cubic Darboux systems with a non-elementary singular point at the Poincar´ e equator. Mathematical notes of NEFU. 2023;30(3):27-37. (In Russ.) https://doi.org/10.25587/SVFU.2023.68.43.004
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