In the twentieth century, with the help of computers and graphical methods, we had a great development of dynamic systems, where the Lorenz equations became well known, showing for the first time that certain systems, due to their great sensitivity to initial conditions, can present an unpredictable response although the models that gave rise to it are well defined and through perfectly deterministic equations. Studies of these systems have greatly advanced the solution of many mechanical problems, now represented by nonlinear models. This new mechanics brought with it a new vision for older problems and a change in the conceptual bases of mechanics itself with phase spaces and phase planes, fractal geometry, etc. In addition to all of this, we have recently had a major advance in nanotechnology, enabling the development of nanorobotics (nanomachines) with revolutionary applications in many fields of knowledge, especially medicine. In this chapter, we will briefly address some of the most important developments in nonlinear mechanics and one presents the Krylov–Bogoliubov-Mitropolsky (KBM) methods for finding an analytical solution for weakly nonlinear systems. Furthermore, we will study the use of the DNA molecule to build nanomachines, including many applications.