On the study of slow–fast dynamics, when the fast process has multiple invariant measures

Benjamin D. Goddard, Michela Ottobre, Kevin J. Painter, Iain Souttar

Research output: Contribution to journalArticlepeer-review


Motivated by applications to mathematical biology, we study the averaging problem for slow–fast systems, in the case in which the fast dynamics is a stochastic process with multiple invariant measures. We consider both the case in which the fast process is decoupled from the slow process and the case in which the two components are fully coupled. We work in the setting in which the slow process evolves according to an ordinary differential equation (ODE) and the fast process is a continuous time Markov process with finite state space and show that, in this setting, the limiting (averaged) dynamics can be described as a random ODE (i.e. an ODE with random coefficients).
Original languageEnglish
Article number20230322
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2278
Early online date11 Oct 2023
Publication statusPublished - Oct 2023


  • averaging
  • collective navigation
  • interacting piecewise deterministic Markov processes
  • multiscale dynamics
  • processes with multiple invariant measures

ASJC Scopus subject areas

  • Engineering(all)
  • Physics and Astronomy(all)
  • Mathematics(all)


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