Maximum number of limit cycles for generalized Kukles differential system

• Autorzy: Melki Houdeifa , Makhlouf Amar

Identyfikatory

DOI

10.1515/jaa-2021-2070

• Języki publikacji: EN

Abstrakty

EN

We apply the averaging theory of first and second order to a class of generalized polynomial Kukles differential systems, which can bifurcate from the periodic orbits of the linear center ẋ = y, ẏ = −x, in order to study the maximum number of limit cycles of these systems.

Słowa kluczowe

EN

limit cycle; averaging theory; Kukles systems

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2023
• Tom: Vol. 29, nr 1
• Strony: 59--75
• Opis fizyczny: Bibliogr. 22 poz., wykr.

Twórcy

autor

Melki Houdeifa

• Department of Mathematics, Laboratory LMA, University of Annaba, P. O. Box 12, Annaba 23000, Algeria

autor

Makhlouf Amar

• Department of Mathematics, Laboratory LMA, University of Annaba, P. O. Box 12, Annaba 23000, Algeria

Bibliografia

• [1] S. Badi, E. Bendib and A. Makhlouf, On the maximum number of limit cycles for a generalization of polynomial Liénard differential systems via averaging theory, J. Pure Appl. Algebra 12 (2016), no. 4, 2971-2985.
• [2] A. Boulfoul, A. Makhlouf and N. Mellahi, On the limit cycles for a class of generalized Kukles differential systems, J. Appl. Anal. Comput. 9 (2019), no. 3, 864-883.
• [3] A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math. 128 (2004), no. 1, 7-22.
• [4] Y. Cao and C. Liu, The estimate of the amplitude of limit cycles of symmetric Liénard systems, J. Differential Equations 262 (2017), no. 3, 2025-2038.
• [5] J. Chavarriga, E. Sáez, I. Szántó and M. Grau, Coexistence of limit cycles and invariant algebraic curves for a Kukles system, Nonlinear Anal. 59 (2004), no. 5, 673-693.
• [6] T. Chen and J. Llibre, Limit cycles of a second-order differential equation, Appl. Math. Lett. 88 (2019), 111-117.
• [7] J. Giné, Conditions for the existence of a center for the Kukles homogeneous systems, Comput. Math. Appl. 43 (2002), no. 10-11, 1261-1269.
• [8] J. Giné, J. Llibre and C. Valls, Centers for the Kukles homogeneous systems with odd degree, Bull. Lond. Math. Soc. 47 (2015), no. 2, 315-324.
• [9] J. Giné, J. Llibre and C. Valls, Centers for the Kukles homogeneous systems with even degree, J. Appl. Anal. Comput. 7 (2017), no. 4, 1534-1548.
• [10] D. Hilbert, Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Congress zu Paris 1900, Nachr. Ges. Wiss. Göttingen Math. Phys. Kl. 5 (1900), 253-297.
• [11] F. Jiang, Z. Ji and Y. Wang, An upper bound for the amplitude of limit cycles of Liénard-type differential systems, Electron. J. Qual. Theory Differ. Equ. 2017 (2017), Paper No. 34.
• [12] I. S. Koukless, Sur quelques cas de distinction entre un foyer et un centre, C. R. Dokl. Acad. Sci. URSS (N. S.) 42 (1944), 208-211.
• [13] J. Llibre, C. A. Buzzi and P. R. d. Silva, 3-dimensional Hopf bifurcation via averaging theory, Discrete Contin. Dyn. Syst. 17 (2007), no. 3, 529-540.
• [14] J. Llibre and A. C. Mereu, Limit cycles for generalized Kukles polynomial differential systems, Nonlinear Anal. 74 (2011), no. 4, 1261-1271.
• [15] J. Llibre, A. C. Mereu and M. A. Teixeira, Limit cycles of the generalized polynomial Liénard differential equations, Math. Proc. Cambridge Philos. Soc. 148 (2010), no. 2, 363-383.
• [16] J. Llibre and C. Valls, Limit cycles for a variant of a generalized Riccati equation, Appl. Math. Lett. 68 (2017), 76-79.
• [17] A. Makhlouf and A. Menaceur, On the limit cycles of a class of generalized Kukles polynomial differential systems via averaging theory, Int. J. Differ. Equ. 2015 (2015), Article ID 325102.
• [18] N. Mellahi, A. Boulfoul and A. Makhlouf, Maximum number of limit cycles for generalized Kukles polynomial differentia systems, Differ. Equ. Dyn. Syst. 27 (2019), no. 4, 493-514.
• [19] A. P. Sadovskiĭ, Cubic systems of nonlinear oscillations with seven limit cycles, Differ. Uravn. 39 (2003), no. 4, 472-481.
• [20] J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Appl. Math. Sci. 59, Springer, New York, 1985.
• [21] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, Berlin, 1990.
• [22] L. Yang and X. Zeng, An upper bound for the amplitude of limit cycles in Liénard systems with symmetry, J. Differential Equations 258 (2015), 2701-2710.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).