Cubic nonlinear differential system, their periodic solutions and bifurcation analysis

<abstract><p>In this article, periodic solutions from a fine focus $ U = 0 $, are accomplished for several classes. Some classes have polynomial coefficients, while the remaining classes $ C_{14, 7} $, $ C_{16, 8} $ and $ C_{5, 5}, $ $ C_{6, 6} $ have non-homogeneous and homogenous trigonometric coefficients accordingly. By adopting a systematic procedure of bifurcation that occurs under perturbation of the coefficients, we have succeeded to find the highest known multiplicity $ 10 $ as an upper bound for the class $ C_{9, 4} $, $ C_{11, 3} $ with algebraic and $ C_{5, 5}, $ $ C_{6, 6} $ with trigonometric coefficients. Polynomials of different degrees with various coefficients have been discussed using symbolic computation in Maple 18. All of the results are executed and validated by using past and present theory, and they were found to be novel and authentic in their respective domains.</p></abstract>

Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

This article concerns with the development of the number of focal values. We analyzed periodic solutions for first-order cubic non-autonomous ordinary differential equations. Bifurcation analysis for periodic solutions from a fine focus {\mathfrak{z}}=0 is also examined. In particular, we are interested to detect the maximum number of periodic solutions for various classes of higher order in which a given solution can bifurcate under perturbation of the coefficients. We calculate the maximum number of periodic solutions for different classes, namely, {C}_{10,5} and {C}_{12,6} with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes {C}_{8,3} and {C}_{8,4} with algebraic coefficients have at most eight limit cycles. The new formula {\varkappa }_{10} is developed by which we succeeded to find highest known multiplicity ten for class {C}_{\mathrm{9,3}} with polynomial coefficient. Periodicity is calculated for both trigonometric and algebraic coefficients. Few examples are also considered to explain the applicability and stability of the methods presented.

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.

Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

Objective Lens

Managing editor Dr. John L. McKillip shares his perspective on this issue of Fine Focus.