DELAYED RANDOM-WALKS

The fluctuations about the stable point in a delayed dynamical system are modeled as a delayed random walk: i.e., a random walk in which the transition probability depends on the position of the walker at a time tau in the past and transitions in the direction of the stable point are more probable. It is shown that, depending on the magnitude of the delay, the root mean square displacement root[X(2)(t)] versus time interval approaches a limiting value in either an oscillatory or nonoscillatory fashion. This limiting value of root[X(2)(t)] is a linear function of tau.