Computing simple invariant solutions in chaotic dynamical systems

One approach to understanding the chaotic dynamics of nonlinear dissipative systems is through their simple invariant solutions, such as steady states and periodic orbits. For high-dimensional systems, including fluid flows, computing these solutions is particularly challenging. In this talk, I will review the methods developed over the past three decades for computing invariant solutions in such high-dimensional systems. While the underlying ideas are simple, implementing these methods involves many subtleties. 

As an example, I will single out multiple shooting for periodic orbits. In this method, more than one point is sought such that the trajectory from each one connects to the next, so that together they form a closed loop. These points satisfy a system of nonlinear equations, which is typically solved with Newton’s method. The linear system for each Newton step can be written in different matrix forms depending on the order in which the equations are stacked. Although these matrix forms encode the same equations, the cost of solving them with an iterative solver such as GMRES varies significantly with the ordering. I will show how this cost scales with the number of points for different orderings and discuss how the scaling depends on the structure of the governing equations.

Venue: Fulton G20