AUTHOR=Akram Saima , Nawaz Allah , Yasmin Nusrat , Ghaffar Abdul , Baleanu Dumitru , Nisar Kottakkaran Sooppy 

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

JOURNAL=Frontiers in Physics

VOLUME=8

YEAR=2020

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.00264

DOI=10.3389/fphy.2020.00264

ISSN=2296-424X

ABSTRACT=<p>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 <italic>z</italic> = 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 <italic>C</italic><sub>3, 8</sub>, <italic>C</italic><sub>4, 3</sub>, <italic>C</italic><sub>7, 5</sub>, <italic>C</italic><sub>7, 6</sub>, eight periodic multiplicities have been found. To investigate the multiplicity &gt;9, the formula for the focal value was not available in the literature. We also succeeded in constructing the formula for η<sub>10</sub>. By implementing our newly developed formula, we are able to get multiplicity ten for classes <italic>C</italic><sub>7, 3</sub>, <italic>C</italic><sub>9, 1</sub>, 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.</p>