Periodic Solutions of Some Classes of One Dimensional Non-autonomous Equation

• Main Authors: Saima Akram, Allah Nawaz, Nusrat Yasmin, Abdul Ghaffar, Dumitru Baleanu, Kottakkaran Sooppy Nisar
• Format: Article
• Language: English
• Published: Frontiers Media S.A. 2020-09-01
• Series: Frontiers in Physics
• Subjects: multiplicity; periodic solution; non-autonomous equation; bifurcation method; trigonometric coefficients
• Online Access: https://www.frontiersin.org/article/10.3389/fphy.2020.00264/full

In this paper, the periodic solutions of a certain one-dimensional differential equation are investigated for the first order cubic non-autonomous equation. The method used here is the bifurcation of periodic solutions from a fine focus z = 0. We aimed to find the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. For classes C3, 8, C4, 3, C7, 5, C7, 6, eight periodic multiplicities have been found. To investigate the multiplicity >9, the formula for the focal value was not available in the literature. We also succeeded in constructing the formula for η10. By implementing our newly developed formula, we are able to get multiplicity ten for classes C7, 3, C9, 1, which is the highest known to date. A perturbation method has been properly established for making the maximal number of limit cycles for each class. Some examples are also presented to show the implementation of the newly developed method. By considering all of these facts, it can be concluded that the presented methods are new, authentic, and novel.