Fractional derivatives appear to be a convenient and effective tool for describing complex processes, systems, and material characteristics. In this study these mathematical operations are used to modify an exemplary non-linear system (the Duffing system) by adding additional fractional components. Numerical analyses were performed to check how the orders of derivatives in fractional terms affect the energy efficiency of the modified system. A numerical differential transform method ($DTM$) was applied to solve fractional-order differential equations quickly and effectively. This analysis focuses on fractional terms with small intensity coefficients and low values of the fractional orders of derivatives (i.e. close to integer values). The results show that in some cases the fractional elements clearly modify the system dynamics and significantly increase the system energy efficiency. Different fractional derivatives can have markedly different qualitative effects and we show that introducing multiple fractional terms can stabilise changes to the energy efficiency with, for example, high efficiency intermittent bistable solutions.

分数导数似乎是描述复杂过程、系统和材料特性的一种方便有效的工具。在本研究中，这些数学运算用于通过添加额外的分数分量来修改示例性非线性系统（Duffing 系统）。进行数值分析以检查分数阶导数如何影响修改后系统的能量效率。一种数值微分变换方法（$D吨米$) 用于快速有效地求解分数阶微分方程。该分析侧重于强度系数小且导数的小数阶值低（即接近整数值）的小数项。结果表明，在某些情况下，分数元素明显改变了系统动力学并显着提高了系统能效。不同的分数导数可能具有明显不同的定性影响，我们表明，引入多个分数项可以稳定能源效率的变化，例如高效间歇双稳态解决方案。