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 . 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 , eight periodic multiplicities have been observed. The new formulas and are constructed. We used our new formulas to find the maximum multiplicity for class . We have succeeded to determine the maximum multiplicity ten for class 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.

中文翻译：

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三次非自治微分方程多重周期解的存在性

本文考虑了估计常微分方程周期解数的方法。讨论了确定周期解的条件。我们通过使用周期分析的分叉分析方法研究了一阶微分非自治方程的焦点值。。尤其要关注希尔伯特第十六个问题的第二部分，我们对检测周期解的最大数量感兴趣，在给定的解中，给定解可以在系数的扰动下分成两部分。对于某些类，已经观察到八个周期性的多重性。新公式和被构建。我们使用新的公式来查找class的最大多重性。我们已成功确定类别的最大多重性10，这是迄今为止可用文献中已知的最高多重性。另一个挑战是检查所讨论方法的适用性，这是通过提供一些示例来实现的。总体而言，所讨论的结果在其研究领域中是新颖，真实和新颖的。