**6. Conclusions**

In this chapter, we describe the applications of atomic force microscopy. We also analyzed the nonlinear dynamic behavior of a mathematical model considering the viscoelasticity term and the microcantilever deflection. In this way, we establish the behavior of the initial conditions and which ones have greater entropy and more fractality. These analyzes corroborate to determine the set of initial conditions for our dynamic analyses, as observed in **Figure 3** and **Table 3**.

After analyzing the behavior of the initial conditions, the dynamic behavior of the dimensionless parameter was analyzed, which considers viscoelasticity and, therefore, the regions in which the system presents a chaotic behavior. This behavior was obtained using the maximum Lyapunov exponent, and for a given set of parameters, it was observed by the bifurcation diagram and the Poincaré map. In this way, the ranges for the parameters *p* and *η* were established where a possible chaotic behavior occurs, as we see in **Figures 5** and **7**.

The results obtained by the analysis of the basins of attraction showed a strong influence between the parameters p and eta in the initial conditions. As we observed in the calculation of entropy and uncertainty coefficient for the grid of initial conditions *x*<sup>0</sup> <sup>1</sup> � *<sup>x</sup>*<sup>0</sup> <sup>2</sup> ¼ �½ �� � 0*:*9,0*:*9 ½ � 0*:*9,0*:*9 , have regions of high fractality and receive it from a new attractor, as shown in **Figures 4a–c** and **5a–c**. Considering the initial condition 0½ � *:*1,0*:*0 the MLE has regions of positive value, that is, there is a chaotic behavior, as shown in **Figure 6**.

These analyses corroborate to determine the *p* and η parameters for the application of two control techniques and suppress the chaotic behavior. This suppression allows us to have a better understanding of the microcantilever when reading biological samples that can generate chaotic movements. These chaotic movements can be detected as noises in the structure of the AFM device.

In general, it is possible to notice that the two control methods presented low errors as shown in **Figure 16**. For this system there was no difference in convergence to the intended orbit; however, it is possible to notice that the OLFC control has a simple implementation methodology in relation to the SDRE, as it excludes

*MEMS-Based Atomic Force Microscope: Nonlinear Dynamics Analysis and Its Control DOI: http://dx.doi.org/10.5772/intechopen.108880*

nonlinearities, facilitating the application of the control method and enabling a possible practical implementation using embedded systems. **Figure 17** showed this comparison between the control techniques.

Due to the low computational cost, the OLFC control technique proves to be a viable alternative for embedded systems of the AFM type. Works such as [34] make a comparison of computational costs.
