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.
Highlights
The bifurcation analysis has attracted the attention of many researchers because it has wild applications in dynamics system and the universal existence of bifurcation in the nature, for example, bifurcation occurs when the small smooth change of parameters leads to qualitative change of its behavior, aerodynamic limit cycle oscillation, and nonlinear oscillation in power system; the homoclinic and heteroclinic branches of a limit cycle colliding with one saddle point, two, or more saddle points and multiple biological dynamical systems are bifurcation.As nature is changing every moment and many changes occurring in nature are periodic like weather, blood flow inside body, circadian rhythm, oceans, and even human behavior, the study of theory of periodic or almost periodic solution is gaining attention
We presented the maximum number of periodic orbits which are usually called limit cycles
Limit cycles are closed paths that are isolated from set of all periodic orbits
Summary
The bifurcation analysis has attracted the attention of many researchers because it has wild applications in dynamics system and the universal existence of bifurcation in the nature, for example, bifurcation occurs when the small smooth change of parameters leads to qualitative change of its behavior (see [1]), aerodynamic limit cycle oscillation, and nonlinear oscillation in power system; the homoclinic and heteroclinic branches of a limit cycle colliding with one saddle point, two, or more saddle points and multiple biological dynamical systems are bifurcation. Neto in [2] states that for equation (1), we are unable to have upper bound for number of periodic solutions until some coefficients are restricted. To find maximum number of periodic solutions, we use complexified form of equation (1). For equation (1), complexified form is used so that we can take the maximum number of periodic solutions for each class using the perturbation method. For this reason, periodic solutions cannot be destroyed by any small perturbation of the coefficients.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
