An analysis of the effect of parallelism in the control of dynamical systems†

Abstract

In certain physical systems measuring the value of one variable, or setting the value of one parameter, of the system alters the values of any number of other variables and parameters unpredictably. We show in this paper that under these conditions a parallel approach succeeds in carrying out the required measurement of variable, or setting of parameter, values while a sequential approach fails. Two dynamical systems are provided as examples of this phenomenon. The short-term dynamical behavior of such systems is important in the context of real-time control applications, where the variables and parameters of a system need to be monitored on a continuous basis and measured and/or set at regular intervals. Thus, in a Belousov–Zhabotinskii chemical reaction, measurement disturbs the equilibrium of the system and causes it to enter into an undesired state. If, however, several measurements are performed in parallel, the effect of perturbations cancels out and the system remains in a stable state. Similarly, the forced damped oscillation model is led into chaotic behavior when its parameters are changed sequentially, but retains its stable behavior when the changes are made in parallel. These results confirm the existence of physical systems with the property that certain operations on them can be performed successfully in parallel but not sequentially. We conclude by describing a potential system that possesses this property in each of the three aspects of computation that it combines, namely, measurement and/or setting of physical quantities, as well as the conventional arithmetic and logical operations on numbers. This research was supported by the Natural Sciences and Engineering Research Council of Canada.