In this work, the nonlinear dynamics and control of an atomic force microscopy (AFM) in fractional-order are investigated. Numerical simulations show the existence of chaotic behavior for some regions in the parameter space, whose behavior is characterized using the power spectral density and the 0-1 test. To bring the system from a chaotic state to a periodic one, the nonlinear saturation control (NLSC) and the time-delayed feedback control (TDFC) techniques for the fractional-order systems are applied with and without accounting for the fractional-order. Numerical results show the influence of fractional-order derivative on the dynamics of the AFM system. Due to that, some phenomena arise, which are confirmed through detailed numerical investigations by the 0-1 test. The NLSC and TDFC techniques showed to be efficient in controlling the chaotic behavior of the AFM in fractional-order.

在这项工作中，研究了分数阶原子力显微镜 (AFM) 的非线性动力学和控制。数值模拟表明参数空间中某些区域存在混沌行为，其行为特征使用功率谱密度和 0-1 检验。为了使系统从混沌状态进入周期性状态，分数阶系统的非线性饱和控制 (NLSC) 和延时反馈控制 (TDFC) 技术在考虑和不考虑分数阶的情况下应用。数值结果显示了分数阶导数对 AFM 系统动力学的影响。因此，出现了一些现象，这些现象通过 0-1 测试的详细数值研究得到证实。