I am also interested on the Second Method of Lyapunov for describing Dynamical Systems. This reseach relies on numerical analysis and PDEs as we attempt to contruct Complete Lyapunov Function for dynamical systems by solving the corresponing PDEs with numerical methods. During my research, I have released (and currently I mantain) two user-friendly freely-distributed open codes to construct complete Lyapunov functions as well as to construct and evaluate any order Wendland function (see my publication list).
Further, I am currently also collaborating with Prof. Michael Melgaard at University of Sussex. Together we are studying the influence of magnetic fields, both externally and internally, for PDE problems arrising within Quantum Mechanics and Quantum Chemistry. This work relies mostly on mathematical analysis and PDEs.