Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

Abstract

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 Z=0. 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 C7,7,C8,5,C8,6,C8,7, eight periodic multiplicities have been observed. The new formulas ξ10 and ϰ10 are constructed. We used our new formulas to find the maximum multiplicity for class C9,2. We have succeeded to determine the maximum multiplicity ten for class C9,2 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.

Periodic Solutions for Two Dimensional Quartic Non-Autonomous Differential Equation

“…Using polynomial coefficient "z" for various higher-order classes, we have calculated the possible maximum number of periodic solutions 9 for class C 4;1 and 8 for the classes C 5;1 , C 6;1 , C 7;1 . The verification of the presented below theorems stems from papers by Alwash et al [7,8] and by Saima et al please see the example, [1][2][3].…”

“…This article contains several recent developments and advances in calculations of periodic solutions and their applications in various areas of the mathematical, physical and engineering sciences. We have investigated upper bounds for the non-autonomous ordinary differential equation (ODE) of the cubic degree [1][2][3]. The primary question striking in our minds is to investigate the maximum number of periodic solutions when the degree of non-autonomous (ODE) is increased from three to four; so, we started working for the quartic system.…”