NONLINEAR DYNAMICS AND CHAOS

This introductory course on dynamical systems is mainly concerned with linear systems, and one and two-dimensional nonlinear systems of differential equations (low-dimensional nonlinear systems of differential equations, and iterations of low-dimensional maps). Students investigate how to determine the qualitative behavior of the solutions of these differential equations, without having to determine the actual solutions explicitly. This is an applied mathematics course; the main focus is on understanding and explaining the behavior of solutions to differential equations, as opposed to a pure mathematics course where the focus might be more on stating and proving theorems. Topics include linear systems of differential equations (linear stability theory) and nonlinear systems (existence and uniqueness, numerical methods, one and two-dimensional flows, phase space, limit cycles, Poincare-Bendixson theorem, bifurcations, Hopf bifurcation, the Lorenz equations, and chaos).