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 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.

On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

Radially Symmetric Solutions of

We investigate solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, , such that as and as . Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies for infinitely many values . At the critical value there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime we prove the existence of a family of “super singular” sign changing singular solutions.

Radially Symmetric Solutions of a Nonlinear Elliptic Equation

We investigate the existence and asymptotic behavior of positive, radially symmetric singular solutions of , . We focus on the parameter regime and where the equation has the closed form, positive singular solution , . Our advance is to develop a technique to efficiently classify the behavior of solutions which are positive on a maximal positive interval . Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the behavior of solutions in the phase plane of the autonomous equation. We then show how specific solutions of the autonomous equation give rise to the existence of several new families of singular solutions of the equation. Specifically, we prove the existence of a family of singular solutions which exist on the entire interval , and which satisfy for all . An important open problem for the nonautonomous equation is presented. Its solution would lead to the existence of a new family of “super singular” solutions which lie entirely above .