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, C 10 , 5 {C}_{10,5} and C 12 , 6 {C}_{12,6} with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes C 8 , 3 {C}_{8,3} and C 8 , 4 {C}_{8,4} with algebraic coefficients have at most eight limit cycles. The new formula ϰ 10 {\varkappa }_{10} is developed by which we succeeded to find highest known multiplicity ten for class C 9,3 {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.

中文翻译：

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计算具有代数和三角系数的一阶非自治方程的焦点值

摘要 本文关注焦点值数量的发展。我们分析了一阶三次非自治常微分方程的周期解。还检查了来自精细焦点 z = 0 {\mathfrak{z}}=0 的周期解的分岔分析。特别是，我们有兴趣检测各种高阶周期解的最大数量，其中给定解在系数扰动下可以分叉。我们用三角系数计算不同类的周期解的最大数量，即 C 10 , 5 {C}_{10,5} 和 C 12 , 6 {C}_{12,6}最多九个和八个多重性。具有代数系数的类C 8 , 3 {C}_{8,3} 和C 8 , 4 {C}_{8,4} 最多具有八个极限环。新公式 ϰ 10 {\varkappa }_{10} 是通过它我们成功地找到具有多项式系数的类 C 9,3 {C}_{\mathrm{9,3}} 的最高已知多重性十。计算三角系数和代数系数的周期性。还考虑了几个例子来解释所提出方法的适用性和稳定性。