Summary
A well-established approach for analyzing the stability of delayed differential equations (DDE) is the semi-discretisation method, which approximates the monodromy operator and the characteristic multipliers through linear mapping. While powerful, it can be computationally expensive and hard to implement. An alternative method, the implicit subspace iteration (ISSI), builds upon the same principles but instead of a linear mapping based on a monodromy matrix, it relies only on numerical simulation. By integrating the initial state over the time period and performing pseudo-inverse calculations, the dominant multipliers can be iteratively computed. The advantage of this approach is its flexibility: any numerical solver can be used. Furthermore, it is applicable to difficult problems involving state-dependent delays (SDD), neutral, non-smooth or stiff systems, as long as the system can be numerically simulated. In our current work, we present the extended ISSI method to handle SDD systems. We compare the results of the ISSI with a linearised solution in a case study related to turning operations.
