Periodic Solutions for Different Classes of Abel’s Type Equation

Saima Akram, Allah Nawaz

Abstract

The maximum number of periodic solutions for the ordinary differential equation is computed in this article. Periodic orbits for first order non-autonomous differential equation of Able's type from a fine focus z=0 are investigated. The research goal of the article is to achieve the maximum possible periodic solutions for two distinct polynomials; algebraic and trigonometric coefficients. The scientific novelty of the article lies in the fact that we have used our newly developed formula ϰ_10. With this formula, we determined the highest known multiplicity 10 for the classes C10,1, C10,2 with algebraic and C8,8, C20,10 with trigonometric coefficients, which is never done before in literature. We used the computer algebra program Maple 18 to handle the complicated and time-consuming integral computations required for calculating the periodic solutions. We used the systematic bifurcation method, which occurred when the coefficients are perturbed. The stability of limit cycles belonging to the classes, as mentioned earlier, is also briefly discussed. For the implementation of theoretical concepts, numerous examples are presented. Consequently, the results presented here are original, authentic, and a valuable contribution to the literature.

Keywords: Abel’s equation, periodic solutions, perturbation method, homogeneous and non-homogeneous trigonometric coefficients.


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References


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