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

May Anne Mata1, Rebecca C Tyson2, Priscilla Greenwood3

  • 1Department of Math, Physics, and Computer Science, University of the Philippines Mindanao, Davao City, Philippines. memata@up.edu.ph.

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Summary

This study analyzes stochastic population dynamics, revealing how fluctuations around periodic limit cycles can be understood similarly to those around fixed points. This provides new insights into power spectral density features.

Keywords:
Disease recurrenceEpidemiologyFloquet theoryLimit cycle with noiseOrnstein–Uhlenbeck processPeriodic forcingSeasonal forcingStochastic fluctuationsSustained oscillations

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Area of Science:

  • Mathematical Biology
  • Stochastic Processes
  • Dynamical Systems

Background:

  • Real populations often display periodic stochastic behavior, modeled as limit cycles driven by seasonal parameter variations.
  • Previous research analyzed power spectral density (PSD) of these fluctuations via autocorrelation functions.

Purpose of the Study:

  • To present an alternative analysis of fluctuations around stochastically perturbed limit cycles.
  • To provide a new theoretical framework for understanding population dynamics with periodic forcing.

Main Methods:

  • Analysis of stochastic differential systems with periodic parametric forcing.
  • Development of a new limit theorem near a Hopf point.
  • Examination of stochastic equations for phase and amplitude.

Main Results:

  • Fluctuations around a limit cycle are analogous to those around a fixed point.
  • Fluctuations can be factorized, revealing combined frequencies of the limit cycle and stochastic perturbation.
  • This factorization explains previously observed PSD features.

Conclusions:

  • The study offers a novel understanding of stochastic fluctuations in periodic population dynamics.
  • The findings provide a simplified framework for analyzing complex time series data.
  • This approach enhances the interpretation of power spectral density in ecological and epidemiological models.