Periodic solutions of a differential perturbed system via the averaging theory and the Melnikov method

Main Article Content

Sana Karfes
Elbahi Hadidi


In this paper, we will study the maximum number of limit cycles of a perturbed differential system with respect to its parameters which appear in the system specially on the degree of the polynomials. For this we will use two methods namely the averaging theory of first order and the method of Melnikov on the same system to provide an upper bound for the number of periodic solutions which can bifurcate from the center with ε = 0. In the end, we will present some numerical examples in order to illustrate the theoretical results given by the averaging theory and Melnikov one.


Download data is not yet available.


Metrics Loading ...

Article Details

How to Cite
Karfes, S. and Hadidi, E. 2023. Periodic solutions of a differential perturbed system via the averaging theory and the Melnikov method. Journal of Innovative Applied Mathematics and Computational Sciences. 3, 1 (Jul. 2023), 49–65. DOI:
Research Articles
Author Biography

Elbahi Hadidi




Mathieu, E. (1868). Mémoire sur le mouvement vibratoire dune membrane de forme elliptique. J. Math. Pures Appl. 13 (1868) , 13.

Chen, T., & Llibre, J. (2019). Limit cycles of a second-order differential equation, 88 14 . Appl. Math. Lett., 88, 111-117.

Berezin , I., & Zhidkv, N. (1964). Computing Methods. Pergamon Press, Oxford.

Buica, A., & Llibre, J. (2004). Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math., 128, 7-22.

Cima , A., Llibre, J., & Teixeira, M. A. (2008). Limit cycles of some polynomial differential system in dimension 2, 3 and 4 via averaging theory. Appl.Anal. , 87, 149-164.

Sáez , E., & Szántó, I. (2012). Bifurcations of limit cycles in Kukles systems of arbitrary degree with invariant ellipse. Appl. Math. Lett. , 25, 1695-1700.

Poincaré,, H. (1881). Mémoire sur les courbes définies par une équation différentielle I, II. J. Math. 14 Pures Appl. 7 (1881), 375-422; 8 (1882), 251-296., 7, 375-422.

Karfes , S., Hadidi, E., & Kerker, M. A. (2022). On the maximum number of limit cycles of a planar differential system. Int. J. Nonlinear Anal.Appl. , 13(1), 1462-1478.

Mellahi, N., Boulfoul , A., & Makhlouf, A. (2019). Maximum Number of Limit Cycles for Generalized Kukles Polynomial Differential Systems. Differ. Equ. Dyn. Syst. , 27, 493-514.

Zwillinger, D. (2014). Table of Integrals. Series and Products, ISBN: 978-0-12-384933-5 .

Llibre, J., Moeckel, R., & Simó, C. (2015). Periodic orbits and Hamiltonian systems. Advanced Courses in Mathematics CRM Barcelona, Birkhäuser.

Hirano , N., & Rybicki, S. (2003). Existence of limit cycles for coupled Van der Pol equations. J. Differ. Equ. , 195(1), 194-209.

Perko, L. (2000). Differential Equations and Dynamical Systems. (م 7). edition.Springer, New York.

Shi, H., Bai, Y., & Han, M. (2020). On the maximum number of limit cycles for a piecewise smooth differential system. Bull. Sci. Math., 163.

Llibre , J., & Mereu, A. C. (2011). Limit cycles for generalized Kukles polynomial differential systems. Nonlinear Anal. , 74, 1261-1271.

Buica, A., Giné , J., & Llibre, J. (2010). Bifurcation of limit cycles from a polynomial degenerate center. Adv. Nonlinear Stud. , 10, 597-609.

Badi, S., & Makhlouf, A. (2011). Limit cycles of the generalized Liénard differential equation via averaging theory. Ann. Of Di . Eqs., 27(4), 472-479.