Random fluctuations around a stable limit cycle in a stochastic system with parametric forcing

  • May Anne MataEmail author
  • Rebecca C. Tyson
  • Priscilla Greenwood


Many real populations exhibit stochastic behaviour that appears to have some periodicity. In terms of populations, these time series can occur as limit cycles that arise through seasonal variation of parameters such as, e.g., disease transmission rate. The general mathematical context is that of a stochastic differential system with periodic parametric forcing whose solution is a stochastically perturbed limit cycle. Earlier work identified the power spectral density (PSD) features of these fluctuations by computation of the autocorrelation function of the stochastic process and its transform. Here, we present an alternative analysis which shows that the structure of the fluctuations around the limit cycle is analogous to that of fluctuations about a fixed point. Furthermore, we show that these fluctuations can be expressed, approximately, as a factorization which reveals the combined frequencies of the limit cycle and the stochastic perturbation. This result, based on a new limit theorem near a Hopf point, yields an understanding of the previously found features of the PSD. Further insights are obtained from the corresponding stochastic equations for phase and amplitude.


Disease recurrence Epidemiology Floquet theory Limit cycle with noise Ornstein–Uhlenbeck process Periodic forcing Seasonal forcing Stochastic fluctuations Sustained oscillations 

Mathematics Subject Classification

60H10 34F05 34F15 



This research was partially funded by NSERC (Discovery Grant program, Application Number RGPIN-2016-05277)) (RT), and by the Institute for Biodiversity, Resilience, and Ecosystem Services at UBC Okanagan. Towards completion of this research, MAM was supported by the Ph.D. Fellowship Program of the University of the Philippines System.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Math, Physics, and Computer ScienceUniversity of the Philippines MindanaoDavao CityPhilippines
  2. 2.CMPS Department (Mathematics), Barber School of Arts and SciencesUniversity of British Columbia OkanaganKelownaCanada
  3. 3.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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