Linearity is only an approximation for real systems, including control systems. In practice, linear methods must be supplemented with various linearization techniques, or brute force dynamic simulations, to understand deviations from linear predictions. However, when the real system is built and measured, there are usually still a few surprising results for the designer involving decreased stability and noisy dynamics. This course will consider nonlinear methods for understanding nonlinear performance: the idea is to use efficient computer techniques that retain the full nonlinearity. We will begin with studying the logistics map, a nonlinear difference equation whose time series solution is elementary, but which nonetheless has broad applicability in biology, economics, business, music, and engineering. We will use this simple nonlinear map to explore phase space, bifurcations, Poincare maps, strange attractors, and characteristic behavior of nonlinear systems. We will then extend these same concepts to nonlinear differential equations for control systems. We will explore the dynamics of a controlled, electrostatically actuated, MEMS cantilever, which includes dynamic multi-stability, Hopf bifurcations, and chaos. Finally, we will look briefly at ideas for exploiting chaos to enhance control.