Resetting and entraining the Radial Isochron Clock (aka Poincare Oscillator)

A simple model of a nonlinear oscilator has an attracting stable, circular limit cycle with a single unstable fixed point and constant angular velocity. Over the years, this model has been proposed by many. I learned about it from Art Winfree. I like this model because it gives insight into dynamics observed experimentally when oscillating heart cells are stimulated with single or periodic stimuli. A pulsatile stimulus will reset the phase of the oscillator, and periodic pulsatile stimuli will lead to stable phase locking, chaos, or quasiperiodicity. If we assume instantaneous return to the limit cycle after a stimulus, the dynamics are during periodic stimulation are described a circle map that depends on two parameters: the amplitude and frequency of the periodic pulses
(Guevara and Glass, 1982; Keener and Glass, 1984).

The question is: how do the locking zones evolve as one changes not only the amplitude and frequency of the stimulation, but also the rate of return to the limit cycle following a perturbation. Except for me and my collaborators, very few other people have been interested in this problem. I will describe the evolution of the phase locking zones as the rate of the return to the limit cycle varies (Glass and Sun, 1994; Langfield, Facanha, Oldeman, Glass 2017).


M.R. Guevara, L. Glass. Phase-locking, period-doubling bifurcations and chaos in a mathematical model of a periodically driven biological oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias. Journal of Mathematical Biology 14, l-23 (1982).

J. Keener, L. Glass. Global bifurcations of a periodically forced nonlinear oscillator. Journal of Mathematical Biology 21, 175-190 (1984).

L. Glass, J. Sun. Periodic forcing of a limit cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations. Physical Review E 50, 5077-5084 (1994).

P. Langfield, W. L. C. Facanha, B. Oldeman, L. Glass. Bifurcations in a periodically stimulated limit cycle oscillator with finite relaxation times. SIAM Journal on Applied Dynamical Systems 16, 1045-1069 (2017).