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Periodic Solutions for Two Dimensional Quartic Non-Autonomous Differential Equation

Saima Akram1,*, Allah Nawaz1, Muhammad Bilal Riaz2, Mariam Rehman3

1 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, 60000, Pakistan
2 Department of Mathematics, University of Management and Technology, Lahore, 54000, Pakistan
3 Department of Information Technology, Government College University, 38000, Faisalabad

* Corresponding Author: Saima Akram. Email: email

(This article belongs to this Special Issue: Recent Trends in Computational Methods for Differential Equations)

Intelligent Automation & Soft Computing 2022, 31(3), 1467-1482.


In this article, the maximum possible numbers of periodic solutions for the quartic differential equation are calculated. In this regard, for the first time in the literature, we developed new formulae to determine the maximum number of periodic solutions greater than eight for the quartic equation. To obtain the maximum number of periodic solutions, we used a systematic procedure of bifurcation analysis. We used computer algebra Maple 18 to solve lengthy calculations that appeared in the formulae of focal values as integrations. The newly developed formulae were applied to a variety of polynomials with algebraic and homogeneous trigonometric coefficients of various degrees. We were able to validate our newly developed formulae by obtaining maximum multiplicity nine in the class C4,1 using algebraic coefficients. Whereas the maximum number of periodic solutions for the classes C4,4; C5,1; C5,5; C6,1; C6:6; C7,1 is eight. Additionally, the stability of limit cycles belonging to the aforementioned classes with algebraic coefficients is briefly discussed. Hence, we conclude from the above-stated facts that our new results are a credible, authentic and pleasant addition to the literature.


Cite This Article

S. Akram, A. Nawaz, M. Bilal Riaz and M. Rehman, "Periodic solutions for two dimensional quartic non-autonomous differential equation," Intelligent Automation & Soft Computing, vol. 31, no.3, pp. 1467–1482, 2022.

cc This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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