Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focus . Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like , eight periodic multiplicities have been observed. The new formulas and are constructed. We used our new formulas to find the maximum multiplicity for class . We have succeeded to determine the maximum multiplicity ten for class which is the highest known multiplicity among the available literature to date. Another challenge is to check the applicability of the methods discussed which is achieved by presenting some examples. Overall, the results discussed are new, authentic, and novel in its domain of research.