Entrainment dynamics of forced hierarchical circadian systems revealed by 2-dimensional maps

Guangyuan Liao, Casey Diekman, Amitabha Bose

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of its most important properties. A new tool, called the entrainment map, was recently introduced to study this process for a single oscillator. Here we generalize the map to study the effects of light-dark forcing in a hierarchical system consisting of a central circadian oscillator that drives a peripheral circadian oscillator. We develop techniques to reduce the higher-dimensional phase space of the coupled system to derive a generalized two-dimensional entrainment map. Determining the nature of various fixed points, together with an understanding of their stable and unstable manifolds, leads to conditions for existence and stability of periodic orbits of the circadian system. We use the map to investigate how various properties of solutions depend on parameters and initial conditions including the time to and direction of entrainment. We show that the concepts of phase advance and phase delay need to be carefully assessed when considering hierarchical systems.

Original languageEnglish (US)
Pages (from-to)2135-2161
Number of pages27
JournalSIAM Journal on Applied Dynamical Systems
Issue number3
StatePublished - 2020

All Science Journal Classification (ASJC) codes

  • Analysis
  • Modeling and Simulation


  • Circadian rhythm
  • Coupled oscillators
  • Limit cycle
  • Poincaré map


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