This course is an introduction to how to model and analyze the behavior of a complex system as it changes in time. The course will introduce and review linear and nonlinear differential equations in one and two dimensions, and the elements of phase space analysis, including fixed points, periodic solutions, and their stability. Students will apply these techniques to some of the most famous nonlinear models from fields ranging from physics to neuroscience to ecology including the Logistic Model, Duffing Oscillator, Hodgkins-Huxley equations,and so on. Finally, students will learn about some of the surprising consequences of nonlinearity, such as fractals, synchronization, and chaos.