Control of dynamics via identical time-lagged stochastic inputs

We investigate the impact of a stochastic forcing, comprised of a sum of time-lagged copies of a single source of noise, on the system dynamics. This type of stochastic forcing could be made artificially, or it could be the result of shared upstream inputs to a system through different channel lengths. By means of a rigorous mathematical framework, we show that such a system is, in fact, equivalent to the classical case of a stochastically-driven dynamical system with time-delayed intrinsic dynamics but without a time lag in the input noise. We also observe a resonancelike effect between the intrinsic period of the oscillation and the time lag of the stochastic forcing, which may be used to determine the intrinsic period of oscillations or the inherent time delay in dynamical systems with oscillatory behavior or delays. As another useful application of imposing time-lagged stochastic forcing, we show that the dynamics of a system can be controlled by changing the time lag of this stochastic forcing, in a fashion similar to the classical case of Pyragas control via delayed feedback. To confirm these results experimentally, we set up a laser diode system with such stochastic inputs, which effectively behaves as a Langevin system. As in the theory, a peak emerged in the autocorrelation function of the output signal that could be tuned by the lag of the stochastic input. Our findings, thus, indicate a new approach for controlling useful instabilities in dynamical systems.