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

TLDR: In this paper, the authors formulate a description of fluctuations about the periodic orbit which allows the relation between the stochastic oscillations in the fixed-point phase and the oscillation in the limit cycle phase to be elucidated.

Abstract: Fluctuations and noise may alter the behaviour 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 allows 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 that of an Ornstein-Uhlenbeck process. Their power spectrum is computed analytically within a van Kampen expansion in the inverse system size. The subsequent comparison with numerical simulations, carried out in two different ways, illustrates the effects that can occur due to diffusion in the longitudinal direction.