Analytical solutions to the fractional Riccati equation are considered in this paper. Solutions to fractional differential equations are expressed in the form of fractional power series in the Caputo algebra. It is demonstrated that solutions to higher-order Riccati fractional equations inherit some solutions from lower-order Riccati equations under special initial conditions. Such nested and fractal-like structure of solutions is investigated by means of analytical fractional differentiation operator techniques and computational experiments.

中文翻译：

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分数阶Riccati方程解析解的分形结构

本文考虑了分数 Riccati 方程的解析解。分数阶微分方程的解在卡普托代数中以分数幂级数的形式表示。证明了在特殊初始条件下高阶Riccati分数方程的解继承了低阶Riccati方程的一些解。通过分析分数微分算子技术和计算实验来研究解的这种嵌套和分形结构。