Dynamical Systems and Chaos

It’s an introduction to a big field in applied math: dynamical sys- tems, a name given to a range of mathematical models for systems that evolve over time. We will study both the case where time evolves in discrete steps (“discrete dynamical sys- tems”) or continuously (“ordinary differential equations”). The fundamental definitions of dynamical systems (including various notion of what makes behavior “chaotic”) involve key ideas from analysis, and for simple systems, we can prove some nice properties of the long term behavior of the system, again using analysis-style ideas. For more complicated systems, proofs are harder to come by, so we will also give an emphasis to computational techniques, with regular computer homework problems in Matlab (no previous experience with Matlab will be assumed)