In this paper, we investigate the mechanism of atomic force microscopy in tapping mode (AFM-TM) under the Casimir and van der Waals (VdW) forces. The dynamic behavior of the system is analyzed through a nonlinear dimensionless mathematical model. Numerical tools as Poincaré maps, Lyapunov exponents, and bifurcation diagrams are accounted for the analysis of the system. With that, the regions in which the system presents chaotic and periodic behaviors are obtained and investigated. Moreover, the fractional calculus is introduced into the mathematical model, employing the Riemann-Liouville kernel discretization in the viscoelastic term of the system. The 0-1 test is implemented to analyze the new dynamics of the system, allowing the identification of the chaotic and periodic regimes of the AFM system. The dynamic results of the conventional (integer derivative) and fractional models reveal the need for the application of control techniques such as Optimum Linear Feedback Control (OLFC), State-Dependent Riccati Equations (SDRE) by using feedback control, and the Time-Delayed Feedback Control. The results of the control techniques are efficient with and without the fractional-order derivative.

中文翻译：

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分数阶攻丝模式原子力显微镜动力学与控制的数值探索性分析

在本文中，我们研究了在卡西米尔（Casimir）和范德华（VdW）力作用下，轻敲模式（AFM-TM）的原子力显微镜的机理。通过非线性无量纲数学模型分析系统的动态行为。庞加莱图，李雅普诺夫指数和分叉图等数值工具被用于系统分析。由此，获得并研究了系统呈现混沌和周期性行为的区域。此外，在系统的粘弹性项下采用Riemann-Liouville核离散化将分数演算引入数学模型。进行0-1测试以分析系统的新动态，从而可以识别AFM系统的混沌和周期性状态。常规（整数导数）和分数模型的动态结果表明，需要应用控制技术，例如，最佳线性反馈控制（OLFC），通过使用反馈控制的状态相关Riccati方程（SDRE）和时滞反馈控制。无论有无分数阶导数，控制技术的结果都是有效的。