The behavior of systems involving state-dependent delays

This paper focusses on the strange nature and qualitative behavior associated with the systems characterized by State Dependent-Delay Differential Equations (SD-DDEs). We consider one of the most simple and innocently looking SD-DDEs x(t) = ±x(t-x(t)). This retarded SD-DDE brings a lot of intricacies. It looks linear but is actually a nonlinear SD-DDE in disguise. It exhibits the phenomenon of bifurcation. Also there is a switch in the stability properties of this system. The type of bifurcation exhibited by the system x(t) = - x(t - x(t)) + μ is transcritical. Furthermore, its Taylor series approximation, based on truncation and partial sums, gives no idea of the response. We show that Taylorization of SD-DDEs, which is ubiquitously used in physics and engineering community, could be misleading. We also perform singular and regular perturbation analyses and derive the solution of the small signal perturbed system in terms of the Lambert-W function. We also demonstrate that the instability of SD-DDE is quite different from that of ODEs. Our simulation results reveal that serious errors may occur when SD-DDEs are approximated either by Taylorization or by constant delay systems.