Abstract
In this paper, we first consider a cubic integrable system under general quadratic perturbations. We then study the Melnikov functions of the perturbed system up to the fourth order. Our studies show that the first four Melnikov functions are sufficient to obtain the center conditions for the perturbed system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing does not apply to this article as no new data were created or analyzed in this study. No datasets were generated or analyzed during the current study.
References
Asheghi, R., Nabavi, A.: Higher order Melnikov functions for studying limit cycles of some perturbed elliptic Hamiltonian vector fields. Qual. Theory Dyn. Syst. 18, 289–313 (2019)
Asheghi, R., Nabavi, A.: The third order melnikov function of a cubic integrable system under quadratic perturbations. Chaos, Solitons and Fractals. 139, 110291 (2020)
Buică, A., Gasull, A., Yang, J.: The third order Melnikov function of a quadratic center under quadratic perturbations. J. Math. Anal. Appl. 331, 443–454 (2007)
Buică, A., Llibre, J.: Limit cycles of a perturbed cubic polynomial differential center. Chaos, Solitons & Fractals. 32, 1059–1069 (2007)
Françoise, J.P.: Successive derivatives of a first return map, application to the study of quadratic vector fields. Ergodic Theory Dyn. Syst. 16, 87–96 (1996)
Gasull, A., Prohens, R., Torregrosa, J.: Bifurcation of limit cycles from a polynomial non-global center. J. Dyn. Diff. Eq. 20, 945–960 (2008)
Iliev, I.D.: The number of limit cycles due to polynomial perturbations of the harmonic oscillator. Math. Proc. Cambridge Phil. Soc. 127, 317–322 (1999)
Karlin, J., Studden, W.J.: T-systems: With Applications in Analysis and Statistics. Pure Appl. Math. Interscience Publishers, NewYork, London, Sidney (1966)
Llibre, J., Pèrez del Rìo, J.S., Rodríguez, J.A.: Averaging analysis of a perturbed quadratic center. Nonlinear Anal. 46, 45–51 (2001)
Llibre, J., \(\grave{{\rm S}}\)wirszcz, G.: On the limit cycles of polynomial vector fields. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18, 203–214 (2011)
Tigan, G.: Using Melnikov functions of any order for studying limit cycles. J. Math. Anal. 448, 409–420 (2017)
Acknowledgements
This work is supported by Isfahan University of Technology (IUT).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Asheghi, R. The center conditions for a perturbed cubic center via the fourth-order Melnikov function. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 189 (2022). https://doi.org/10.1007/s13398-022-01333-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-022-01333-2