Abstract
AbstractThis 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.
Highlights
A periodic solution is a solution that periodically depends on the independent variable t
In the entire twentieth century, the investigation of periodic solutions of non-autonomous ordinary differential equations (ODEs) has been profoundly influential in the creation and development of fundamental parts of present mathematics, such as functional analysis, algebraic topology, variational methods, and symplectic techniques (Poincare–Birkhoff-type fixed point theorems, etc.)
Let C10,5 be the class for equation (3), if the coefficients are as follows: δ(s) =(cos2 s + sin2 s), In Section 4.2, we have considered the non-homogeneous trigonometric coefficients for different classes and periodic solutions are calculated
Summary
A periodic solution is a solution that periodically depends on the independent variable t. Most of the previous studies on special DEs have been based on models from mechanics and electronics fields that are far from being totally understood but the recent applications to biology, demography, and economy will introduce new classes of DEs and systems with periodic time dependence, requiring the use of new analytical and topological tools In this regard, the study of periodic solutions of non-autonomous DE is of increasing significance, see refs. With the condition that p0(s) = 1 and p1(s),..., pn(s) are real valued periodic continuous functions This class of equation has received more attention in the literature. Complexified form of equation (1) is used to find the maximum number of periodic solutions, see refs.
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