Ordinary Differential Equations and Dynamical Systems

The course will develop both theory and phenomenology of ordinary differential equations and dynamical systems. As we progress from one-dimensional to two-dimensional flows, to higher dimensional flows exhibiting chaotic behavior, we will cover basic existence theorems and topics such as bifurcations, synchronization of oscillating systems, limit cycles, relaxation oscillators and other nonlinear oscillatory phenomena with applications in the sciences. We will cover basic topics in chaotic systems, including the Lorenz flow, strange attractors, and fractal dimension. If time permits, we will discuss the renormalization group approach to the period doubling cascade.