Unstable dynamical systems: Delays, noise and control

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 − τ have opposite signs. These observations support the utility of control strategies using pulsatile stimuli triggered only when variables exceed certain thresholds.

A Team Approach to Undergraduate Research in Biomathematics: Balance Control

“…A sharpening in the distribution of the fluctuations in the COP is also seen for postural sway during quiet standing when the Achilles' tendons are subjected to low frequency, low amplitude periodic vibration (Figures 3b and d). These observations support the concept that human balance control is maintained by a simple "drift and act" mechanism [21,35,36,37,38,53]. This mechanism proposes that the basin of attraction for the stabilized upright position is small enough so that escapes ("falls") are possible [54,63].…”

Human Balance Control: Dead Zones, Intermittency, and Micro-chaos

“…The first is micro-chaos [22,23,24,36] discussed in Section 2.2. The second relies on the observation that the interplay between time delays and noisy perturbations can transiently stabilize an unstable fixed-point [12,14,15,68,71]. The third strategy is a nonlinear-type of control mechanism which relies on the properties of a saddle point [10,4].…”

“…The presence of a dead zone makes it possible to use a drift and act strategy, namely, energy consuming corrective actions are taken only when the controlled variable exceeds a threshold [68,69,71]. The potential reduction in energy requirements stems from the fact that active control is not continuously required.…”

“…Two general types of explanations have been advanced. First, it is possible that the intermittency may be event-driven, namely corrective forces are generated whenever the controlled variable crosses a threshold [4,10,12,33,34,39,68,69,71]. A well known example of threshold crossing in human balance control is the "safety-net" character of the ankle-hip-step control strategy used by humans to maintain balance in the face of increasingly large perturbations [18,86].…”

Balancing with positive feedback: the case for discontinuous control

“…Thus, there exist two pathways of escape from the origin at xZ0: a relatively slow path and a faster one. This bimodal nature of the distribution in L is related to the number of delayed zero crossings, since if x (t) and x (tKt) have different signs, then there is an increased probability that the evolution of (4.1) will change direction, thus further increasing the time until threshold crossing (Milton et al 2008). In other words, the slower paths arise because the solution is temporarily confined near the origin as reported previously in delay differential equations in the setting of a saddle point (Pakdaman et al 1998;Grotta-Ragazzo et al 1999).…”