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=Volume 8 - 2020

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=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 solution into which a given solution can bifurcate under perturbation of the coefficients. For classes $C_{3,8},C_{4,3},C_{7,5},C_{7,6}$ eight periodic multiplicity have been found. To investigate the multiplicity greater than 9, the formula for focal value was not available in the literature. We also succeeded to construct the formula for $\eta _{10}$, by implementing our newly developed formula we are able to get multiplicity ten for classes $ C_{7,3},C_{9,1}$, which is highest known to date. Perturbation method has been properly established in making the maximal number of limit cycles for each class. To check the implementation of the newly developed method, some examples are also presented. By considering all these facts it can be concluded that presented methods are new, authentic and novel.