Unstable dynamical systems: Delays, noise and control

¹  Joint Science Department, The Claremont Colleges - Claremont, CA 91711, USA
²  Centro de Física I. V. I. C. - Caracas 1020-A, Venezuela
³  Sony Computer Science Laboratories, Inc. - Tokyo, 141-0022, Japan

Corresponding author: jmilton@jsd.claremont.edu

Received: 4 November 2007
Accepted: 30 June 2008

Abstract

Escape from an unstable fixed point in a time-delayed dynamical system in the presence of additive white noise depends on both the magnitude of the time delay, τ, and the initial function. In particular, the longer the delay the smaller the variance and hence the slower the rate of escape. Numerical simulations demonstrate that the distribution of first passage times is bimodal, the longest first passage times are associated with those initial functions that cause the greatest number of delayed zero crossings, i.e. instances where the deviations of the controlled variable from the fixed point at times t and t- have opposite signs. These observations support the utility of control strategies using pulsatile stimuli triggered only when variables exceed certain thresholds.