Uncertainty Principle for Control of Ensembles of Oscillators Driven by Common Noise

TL;DR

This paper explores the fundamental trade-off between reliability and coherence in noisy oscillators, establishing an uncertainty principle that constrains simultaneous control of these properties in oscillator ensembles.

Contribution

It introduces an uncertainty principle linking reliability and coherence in noisy oscillators and analyzes its implications for ensembles with common noise or coupling.

Findings

The phase diffusion constant and Lyapunov exponent can be independently controlled.

The ratio of coherence to reliability remains constant under linear feedback control.

A significant difference exists between identical and non-identical oscillators regarding control effects.

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 multiple 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.