Unraveling multiple 1:1 entrainment regions in the Arnold onion diagram: A study of the circadian Novak–Tyson model

The entrainment of biological oscillators is a fundamental problem in studying dynamical systems and synchronization. The Arnold onion diagram is a key tool for visualizing entrainment patterns in a two-dimensional parameter space, defined by period ($T$) and photoperiod ($\chi$). This paper investigates the entrainment behavior of various oscillatory regimes in the Novak–Tyson (NT) model. While previous studies have documented the presence of Arnold onions featuring a single 1:1 entrainment region, our work introduces the novel emergence of multiple disconnected 1:1 entrainment regions within these diagrams. Through the analysis of dynamical systems, we show that multiple Arnold onions emerge for an unforced system near the Hopf bifurcation, which behaves as a damped oscillator. These findings offer new insights into the complex mechanisms underlying circadian seasonality and its dependence on intrinsic oscillator dynamics.