Eur. Phys. J. Special Topics 223, 677-685 (2014)
https://doi.org/10.1140/epjst/e2014-02133-y

Regular Article

Uncertainty principle for control of ensembles of oscillators driven by common noise

¹ Institute of Continuous Media Mechanics, UB RAS, Perm 614013, Russia
² Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK

Received: 3 March 2014
Revised: 18 March 2014
Published online: 28 April 2014

Abstract

We discuss control techniques for noisy self-sustained oscillators with a focus on reliability, stability of the response to noisy driving, and oscillation coherence understood in the sense of constancy of oscillation frequency. For any kind of linear feedback control – single and recursive delay feedback, linear frequency filter, etc. – the phase diffusion constant, quantifying coherence, and the Lyapunov exponent, quantifying reliability, can be efficiently controlled but their ratio remains constant. Thus, an “uncertainty principle” can be formulated: the loss of reliability occurs when coherence is enhanced and, vice versa, coherence is weakened when reliability is enhanced. Treatment of this principle for ensembles of oscillators synchronized by common noise or global coupling reveals a substantial difference between the cases of slightly non-identical oscillators and identical ones with intrinsic noise.