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

Abstract

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 z=0{\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, C10,5{C}_{10,5} and C12,6{C}_{12,6} with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes C8,3{C}_{8,3} and C8,4{C}_{8,4} with algebraic coefficients have at most eight limit cycles. The new formula ϰ10{\varkappa }_{10} is developed by which we succeeded to find highest known multiplicity ten for class C9,3{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.

Keywords

• focal values
• periodic coefficients
• bifurcation
• non-autonomous equation