**1. Introduction**

18 Nonlinearity, Bifurcation and Chaos - Theory and Applications

[10] Fossas E. & Zinober A., *Adaptive tracking control of nonlinear power converters* In Proceedings IFAC Workshop on Adaptation in Control and Signal Processing.

[12] Mohan N., Undeland T. & Robbins W., "Power Electronics: Converters, applications

[13] Taborda J.A., "Bifurcation analysis in second-order systems with PWM based on zero average dynamics (ZAD) strategy.", *Master Thesis. National University of Colombia (in*

[14] Taborda J.A., Angulo F. & Olivar G., "Smooth Bifurcations in 3D-parameter Space of Digital-PWM Switched Converter", In Proceedings of 2nd IEEE Latin American

symposium on circuits and Systems - LASCAS 2011. Bogotá. Colombia, 2011. [15] Taborda J.A., Angulo F. & Olivar G., "Characterization of chaotic attractors inside band-merging scenario in ZAD-controlled buck converter", Special Issue of 2010

DDays, *International Journal of Bifurcations and Chaos*, (to appear) .

[11] Hart D.,"Power Electronics". Prentice Hall, Madrid, España, 2001.

Connobio. Italy. pp 264-266, 2001.

*Spanish)*, Manizales, Colombia, 2006.

and design", J. Wiley, 1995.

It is known that in Brazil and in the whole world, the development of mathematical modeling techniques for the study of problems involving new technologies and the design of nonlinear structures is an emerging area in Engineering Science. To background to the general problems of non-linear dynamics, see the comprehensive monographs: (Awrejcewicz, 1991) and (Awrejcewicz and Lamarque, 2003).

This chapter addresses this issue, involving the dynamical behavior of Electrical Mechanical Systems (EMS), that are systems responsible for electromechanical energy conversion. It is common knowledge that an electromechanical system can be classified in three different groups, according to scales dimensions: *i*) Macro-Systems >1mm, *ii*) Micro-Systems <1mm and >0.1μm, and *iii*) Nano-Systems <0.1 μm and >0.1nm.

The Macro-Systems are characterized by being visible without using microscope and they are present in our Day to Day activities. The Micro-Systems are generally constructed on silicon chips, and are employed in manufacturing integrated electronic circuits. The Nano-Systems can only be seen and manipulated with the use of the Atomic Force Microscope.

In this research field - MEMS and NEMS -, the researches are based on the fact that, in opposition to traditional macro-scale problems, the atomic forces effects and surface phenomena are preponderant over the forces of inertia and gravity. It is also known that the manufacture of these new products have been based on the binomial consuming time and an enormous amount of financial resources. Thus, it becomes necessary some investment in an interface, which addresses the design and manufacturing that, gives designers of

"MEMS" and "NEMS" the proper tools of mathematical modeling and simulations as well as qualitative dynamic analysis. It should be considering a macro-model, which agrees with the simulation results on a physical level, and with the experimental results, obtained from test structures in a lab. Many surveys have been developed in major research centers in Brazil and throughout the world in this direction. It is announced that several research groups in Brazil are involved with this theme, but most of the research carried out or in development focuses on experimentation or in the use of the numerical method - called finite element method. It is known that the nonlinear phenomena are prevalent in MEMS and NEMS, which is a strong motivation for the work of our research group.

On an Overview of Nonlinear and Chaotic Behavior and

Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 47

 **Figure 1.** AFM photo (source: Bowen and Hilal 2009)

**Figure 2.** (source: Bowen and Hilal 2009)

**Figure 3.** Control of the positioning of the scanner. Adapted from : (Cidade et al. 2003)

The main idea of this chapter lies in the study of "State of the Art" of non-linear dynamic models and the design of their controls, since this new area of research is of paramount importance. This chapter deals with two boundary problems belonging to this line of research, in development at the university of Rio Claro and Bauru UNESP, for us, through research groups and development projects for Graduate (Masters) and Post–Doctor. Mainly for the Atomic Force Microscope.
