Delay systems appear in various contexts, from control theory to the modelling of spatially extended systems. Their stability properties play an important role for the characterization of the dynamical behavior and the response to perturbations. This talk will focus on the Lyapunov spectra and the corresponding Lyapunov vectors occurring in different dynamical regimes of delay systems, in theory and experimental feedback applications. For the simplest case of a steady state, a general solution for its stability is provided in terms of the Lambert function. Several properties and limit cases of this function are shown and related to phenomena in different fields like time-delayed feedback control of periodic orbits [1], anticipating synchronization [2], strong and weak instabilities in feedback systems [3] and master stability functions in delay-coupled networks [4]. The limitations of this far-ranging simple concept are discussed for chaotic dynamics, in which qualitative deviations of the steady state or periodic cases can be analyzed and explained in terms of a stochastic model.
[1] W. Just et al., Phys. Rev. Lett. 78, 203 (1997) [2] H. U. Voss, Phys. Rev. E 61, 5115 (2000) [3] S. Yanchuk and P. Perlikowski, Phys. Rev. E 79, 046221 (2009) [4] A. Englert et al., Phys. Rev. E 83, 046222 (2011)