2008/05/12 by Richard P. Boland, Tobias Galla, Alan J. McKane
Biochemistry, Genetics and Molecular Biology · Computer Science · Environmental Science · Mathematics · Physics and Astronomy · #Computer science #Diffusion #Ecosystem dynamics and resilience #Fixed point #Inverse #Limit (mathematics) #Limit cycle #Mathematical analysis #Mathematics #Noise (video) #Nonlinear Dynamics and Pattern Formation #Physics #Quantum mechanics #Spectral density #Statistical physics #Stochastic process #Transverse plane #cond-mat.stat-mech #nlin.AO #q-bio.QM #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/1742-5468/2008/09/p09001
15 pages, 14 figures
arxiv created 2008/05/12 · openalex publication_date 2008/09/01 · arxiv updated 2009/12/01 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/06
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. © 2008 IOP Publishing Ltd.