Though stable, these orbits cannot be computed as solutions of initial value problems. This project focusses on algorithms for periodic orbits and on studying dynamics in systems with multiple time scales.
Periodic orbits are fundamental objects within dynamical systems.
Attracting periodic orbits can usually be found by solving initial value problems for long times, but it is desirable to compute periodic orbits directly. I have been constructing a new family of algorithms that employ automatic differentiation to compute orbits directly with very high accuracy. In addition, I seek to automatically compute bifurcations of periodic orbits and generate rigorous computer proofs of the qualitative properties of numerically computed dynamical systems. Multiple time scales present challenges in the simulation of dynamical systems. My research is directed at extending the qualitative theory of dynamical systems to apply to systems with multiple time scales and to develop algorithms that compute objects in systems with multiple time scales.