Geodesic
Stable hyperbolic limit cycles for a class of differential systems
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2021), pp. 49-60.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we introduce an explicit expression of invariant algebraic curves for a class of polynomial differential systems, then we introduce an explicit expression of its first integral. Moreover, we determine sufficient conditions for these systems to possess a limit cycle, which can be expressed by an explicit formula. Concrete examples exhibiting the applicability of our results are introduced.
Keywords: Hilbert 16th problem, dynamical system, limit cycle, invariant algebraic curve, first integral. 
@article{IVM_2021_9_a5,
     author = {S. E. Hamizi and R. Boukoucha},
     title = {Stable hyperbolic limit cycles for a class of differential systems},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {49--60},
     publisher = {mathdoc},
     number = {9},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2021_9_a5/}
}
TY  - JOUR
AU  - S. E. Hamizi
AU  - R. Boukoucha
TI  - Stable hyperbolic limit cycles for a class of differential systems
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2021
SP  - 49
EP  - 60
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2021_9_a5/
LA  - ru
ID  - IVM_2021_9_a5
ER  - 
%0 Journal Article
%A S. E. Hamizi
%A R. Boukoucha
%T Stable hyperbolic limit cycles for a class of differential systems
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2021
%P 49-60
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2021_9_a5/
%G ru
%F IVM_2021_9_a5
S. E. Hamizi; R. Boukoucha. Stable hyperbolic limit cycles for a class of differential systems. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2021), pp. 49-60. http://geodesic.mathdoc.fr/item/IVM_2021_9_a5/

[1] Hilbert D., “Mathematische Probleme”, Lecture, Second Internat. Congr. Math. (Paris, 1900), Nachr. Ges. Wiss. Gttingen Math. Phys. Kl., 1900, 253–297 | Zbl

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

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

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

[5] Giné J., Grau M., “Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations”, Nonlinearity, 19 (2006), 1939–1950 | DOI | Zbl

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

[7] Perko L., Differential Equations and Dynamical Systems, Texts in Appl. Math., 7, 3rd edition, Springer-Verlag, New York, 2001 | DOI | Zbl

[8] Bendjeddou A., Cheurfa R., “On the exact limit cycle for some class of planar differential systems”, Nonlinear Diff. Equat. Appl., 14 (2007), 491–498 | DOI | Zbl

[9] Llibre J., Zhao Y., “Algebraic Limit Cycles in Polynomial Systems of Differential Equations”, J. Phys. A: Math. Theor., 40 (2007), 14207–14222 | DOI | Zbl

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

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

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