Geodesic
A class of planar differential systems with explicit expression for two limit cycles
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1588-1597.

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The existence of limit cycles is interesting and very important in applications. It is a key to understand the dynamic of polynomial differential systems. The aim of this paper is to investigate a class of a multi-parameter planar polynomial differential systems. Under some suitable conditions, the existence of two limit cycles, these limit cycles are explicitly given. Some examples are presented in order to illustrate the applicability of our results. algebras.
Keywords: limit cycle, invariant algebraic curve, first integral.
Mots-clés : Riccati equation 
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Saad Eddine Hamizi; Rachid Boukoucha. A class of planar differential systems with explicit expression for two limit cycles. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1588-1597. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a100/

[1] A. Bendjeddou, R. Cheurfa, “On the exact limit cycle for some class of planar differential systems”, Nonlinear differ. equ. appl., 14 (2007), 491–498 | DOI | MR | Zbl

[2] A. Bendjeddou, R. Cheurfa, “Coexistence of algebraic and non-algebraic limit cycles for quintic polynomial differential systems”, Elect. J. of Diff. Equ., 2017:71 (2017), 1–7 | MR | Zbl

[3] R. Benterki, J. Llibre, “Polynomial differential systems with explicit non-algebraic limit cycles”, Elect. J. of Diff. Equ., 2012:78 (2012), 1–6 | MR | Zbl

[4] R. Boukoucha, “Explicit limit cycles of a family of polynomial differential systems”, Elect. J. of Diff. Equ., 2017:217 (2017), 1–7 | MR | Zbl

[5] R. Boukoucha, A. Bendjeddou, “On the dynamics of a class of rational Kolmogorov systems”, J. Nonlinear Math. Phys., 23:1 (2016), 21–27 | DOI | MR | Zbl

[6] R. Boukoucha, “Explicit expression for a first integral for some classes of two-dimensional differential systems”, Siberian Electronic Mathematical Reports, 14 (2017), 903–913 | MR | Zbl

[7] J. Chavarriga, J. Llibre, “Invariant algebraic curves and rational first integrals for planar polynomial vector fields”, J. differential equations, 169:1 (2001), 1–16 | DOI | MR | Zbl

[8] F. Dumortier, J. Llibre, J. Artés, Qualitative Theory of Planar Differential Systems, Universitex, Springer, Berlin, 2006 | MR | Zbl

[9] A. Gasull, H. Giacomini, J. Torregrosa, “Explicit non-algebraic limit cycles for polynomial systems”, J. Comput. Appl. Math., 200 (2007), 448–457 | DOI | MR | Zbl

[10] J. Gin\'{e, “Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations}. Nonlinearity”, M. Grau, 19 (2006), 1939–1950 | MR | Zbl

[11] Bull. Amer. Math. Soc., 8 (1902), 437–479 | DOI | MR | Zbl | Zbl

[12] J. Llibre, Y. Zhao, “Algebraic limit cycles in polynomial systems of differential equations”, J. Phys. A: Math. Theor., 40:47 (2007), 14207–14222 | DOI | MR | Zbl

[13] K. Odani, “The limit cycle of the van der Pol equation is not algebraic”, J. of Diff. Equ, 115 (1995), 146–152 | DOI | MR | Zbl

[14] L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7, Third edition, Springer-Verlag, New York, 2001 | DOI | MR | Zbl

[15] D. Schlomiuk, “Algebraic particular integrals, integrability and the problem of center”, Trans. Amer. Math. Soc., 338:2 (1993), 799–841 | DOI | MR | Zbl

[16] M. Yahiaoui, R. Boukoucha, “Invariant algebraic curves and the first integral for a class of Kolmogorov systems”, Nonlinear studies, 27:1 (2020), 205–212 | MR