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This work presents the investigation of the nonlinear dynamics of an Atomic Force Microscope micro cantilever operating in tapping mode. The mathematical model considers the influence of damping, nonlinear stiffness term and a squeeze film damping in fractional-order. The usage of the fractional damping showed the influence of the order of the derivative on the dynamics of the system (regular or non-periodic) behaviour. A good tool to characterize the fractional-order differential systems is the “0-1 Test”, which qualifies these behaviours. To supress the chaotic motion presented in the AFM, the SDRE control technique is proposed for the case of fractionalorder damping.