Differential equations describe an incredible variety of physical phenomena, from the motion of celestial bodies to the spread of an infectious disease to the time and place of a hurricane's landfall. Dynamical systems theory tries to understand the qualitative features of not one, but all the solutions of a differential equation. It turns out that the set of solutions is organized by certain landmarks like equilibria, periodic orbits, and more generally by invariant manifolds. Understanding these landmarks and how they fit together explains what kinds of behaviors a system can exhibit and the occurrence of transitions between behaviors.

I'll discuss my work on computational methods for studying dynamical landmarks in problems involving ordinary, partial, and delay differential equations. An important theme is that invariant objects can often be reformulated as solutions of functional equations, and that these equations are amenable to numerical and nonlinear analysis. I'll also indicate how the results of careful numerical calculations are combined with fixed point arguments to formulate mathematically rigorous computer assisted proofs. These ideas will be illustrated in some applications like gravitational N-body problems and physiological feedback systems.