On Computation of Lyapunov Exponents by QR Methods with Error Control for Semi-Linear DAEs

In this paper, we propose and analyze numerical methods for computing Lyapunov exponents of semi-linear differential-algebraic equations (DAEs), leveraging smooth QR factorizations and Runge-Kutta (RK) methods with error control and automatic step size selection. We demonstrate how both discrete and continuous QR approaches efficiently approximate Lyapunov exponents by simultaneously solving semi-linear DAEs and their linearized counterparts. The paper details the underlying algorithms, error analysis, and numerical integration techniques, focusing on half-explicit RK (HERK) and explicit singly diagonal implicit RK (ESDIRK) methods. We also provide implementation details and present numerical experiments to illustrate the efficiency of these methods.