Time-delay Systems with Band-limited Feedback

Fast nonlinear devices with time-delayed feedback, developed for applications such as communications and ranging, typically include components that are ACcoupled, i.e. components that block zero frequencies. As an example of such a system, we describe a new opto-electronic device with band-limited feedback that uses a Mach-Zehnder interferometer as passive nonlinearity and a semiconductor laser as a current-to-opticalfrequency converter. Our implementation of the device produces oscillations in the frequency range of tens to hundreds of MHz. We observe periodic oscillations created through a Hopf bifurcation as well as quasiperiodic and high dimensional chaotic oscillations. Motivated by the experimental results, we investigate the steady-state solution and it’s bifurcations in time-delay systems with band-limited feedback and arbitrary nonlinearity. We show that the steady state loses stability, generically, through a Hopf bifurcation, which can be either supercritical or subcritical. As a result of this investigation, we find that band-limited feedback introduces practical advantages, such as the ability to control the characteristic time-scale of the dynamics, and that it introduces differences to Ikeda-type systems already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental “period-2” mode found in Ikeda-type systems.