Delayed Random Walk with a Repulsive Origin
Abstract
An important question concerns how unstable equilibria can be stabilized using timedelayed corrections in the presence of noise. Recently the interplay between noise and delay has been emphasized as an essential component of the neural control of stick balancing at the fingertip [1]: The beneficial effects of noise were related to onoff intermittency; However, the effects of the delay on control were not fully investigated. Here we investigate the effects of delayed corrective movements on the stabilization of an unstable equilibrium using a simpler model, namely a delayed random walk[2]. A delayed random walk for an unstable equilibrium is one in which the transition probability depends on the position of the walker at time tau in the past and transitions in the direction away from the unstable point are more probable. Numerical simulations demonstrate t hat both the first passage time to cross a specified threshold and the return time distribution to the origin increase as the delay increases. Thus the diffusion of the walker away from the unstable fixed point is slower the longer the delay. This is in contrast to the case with the stable fixed point where delay makes diffusion faster. These observations may have implications for the design of artificial and natural feedback control systems in general. [1] J. L. Cabrera and J. G. Milton, "OnOff Intermittency in a Human Balancing", Phys.Rev.Lett. Vol. 89, p.. 158702, 2002. [2] T. Ohira and J. G. Milton, "Delayed Random Walks", Phys.Rev.E Vol. 52, pp. 32773280, 1995.
 Publication:

APS March Meeting Abstracts
 Pub Date:
 March 2004
 Bibcode:
 2004APS..MAR.N9001H