Periodic Solutions for Two Dimensional Quartic Non-Autonomous Differential Equation

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 C-4,C-1 using algebraic coefficients. Whereas the maximum number of periodic solutions for the classes C-4,C-4; C-5,C-1; C-5,C-5; C-6,C-1; C-6:6; C-7,C-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.

Automated summary

In this article, the maximum possible numbers of periodic solutions for the quartic differential equation are calculated, and new formulae for determining these maximum numbers are developed. Through a systematic procedure of bifurcation analysis and using computer algebra software, the authors successfully obtain a series of results, which are validated and discussed. These new results are a valuable addition to the existing literature.