Limit cycles, complex Floquet multipliers and intrinsic noise

TL;DR

This paper investigates how intrinsic noise influences chemical systems with limit cycles, revealing that complex Floquet multipliers can lead to sustained oscillations and increased amplitude near instability boundaries.

Contribution

It demonstrates that intrinsic noise can induce sustained oscillations in systems with limit cycles when Floquet multipliers are complex, extending previous findings from fixed points.

Findings

Intrinsic noise can sustain oscillations in limit cycle systems.

Complex Floquet multipliers lead to increased oscillation amplitude.

The effect is confirmed in multiple model systems through analytical and numerical methods.

Abstract

We study the effects of intrinsic noise on chemical reaction systems, which in the deterministic limit approach a limit cycle in an oscillatory manner. Previous studies of systems with an oscillatory approach to a fixed point have shown that the noise can transform the oscillatory decay into sustained coherent oscillations with a large amplitude. We show that a similar effect occurs when the stable attractors are limit cycles. We compute the correlation functions and spectral properties of the fluctuations in suitably co-moving Frenet frames for several model systems including driven and coupled Brusselators, and the Willamowski-Roessler system. Analytical results are confirmed convincingly in numerical simulations. The effect is quite general, and occurs whenever the Floquet multipliers governing the stability of the limit cycle are complex, with the amplitude of the oscillations increasing as the instability boundary is approached.