Projection of Flows: Dual Basis

This lecture discusses the projection of flow evolution on stable and unstable manifolds, exploring the use of Gram-Schmidt and the concept of dual basis. It covers the definition and application of dual basis, finding dual vectors, and the adjoint operator of the Navier-Stokes equations. The implementation of computing dual basis, handling travelling waves, and quantifying errors in the projection process are also explained. An example application demonstrates the projection of an optimal seed on the unstable manifold using the dual basis.