How limit cycles and quasi-cycles are related in systems with intrinsic noise

TL;DR

This paper analyzes how intrinsic noise influences oscillations in systems with limit cycles, providing a theoretical framework that connects stochastic fluctuations in fixed points and limit cycles, with analytical results validated by simulations.

Contribution

It extends the theoretical understanding of stochastic effects in systems with limit cycles, introducing a method to analyze fluctuations around periodic orbits and their spectral properties.

Findings

Fluctuations transverse to the limit cycle behave like an Ornstein-Uhlenbeck process.

Longitudinal fluctuations are diffusive and analyzed via van Kampen expansion.

Analytical results match numerical simulations, highlighting diffusion effects.

Abstract

Fluctuations and noise may alter the behavior of dynamical systems considerably. For example, oscillations may be sustained by demographic fluctuations in biological systems where a stable fixed point is found in the absence of noise. We here extend the theoretical analysis of such stochastic effects to models which have a limit cycle for some range of the model parameters. We formulate a description of fluctuations about the periodic orbit which allows the relation between the stochastic oscillations in the fixed point phase and the oscillations in the limit cycle phase to be elucidated. In the case of the limit cycle, a suitable transformation into a co-moving frame allow fluctuations transverse and longitudinal with respect to the limit cycle to be effectively decoupled. While longitudinal fluctuations are of a diffusive nature, those in the transverse direction follow a stochastic path more akin to an Ornstein-Uhlenbeck process. Their power spectrum is computed analytically within a van Kampen expansion in the inverse system size. This is carried out in two different ways, and the subsequent comparison with numerical simulations illustrates the effects that can occur due to diffusion in the longitudinal direction.