Nonlinear systems like nerves, heart cells or Belouzov-Zhabotinsky reaction, are often examined by applying periodic stimulation, and computing the rotation number r, which measures the ratio of the number of responses of the stimulated system to the number of stimulations. For the case where these systems are in their oscillatory mode, and the stimulations are relatively weak, there exists a comprehensive theory. In this case it is well known that when plotting the rotation number r vs. the period of the stimulation T, a complete devil's staircase is obtained. In addition to the oscillatory mode, these systems usually have also an excitable mode. Experimental and numerical researches show that for the excitable mode of these systems, periodic stimulation leads to a partial devil's staircase.
We are trying to understand this phenomenon of partial devil's staircases through one dimensional maps of the form:
where f(x) has a "jump down" discontinuity. This type of maps arises from (generalized) phase response curves. It is known that when f(x) is monotonically increasing everywhere other than in the discontinuity, it produces complete devil's staircase. We have shown that partial devil's staircases, like those seen in experimental and numerical works, are linked with a non increasing part of the map, on one side of the discontinuity point. Now we are trying to understand the linkage between periodicly stimulated excitable systems, and this type of maps.
Properties
and features of asymmetric partial devil's
staircases deduced from piecewise linear maps
E. Yellin and A. Rabinovitch,
Phys. Rev. E 67, 016202 (2003)
I hope to justify soon the letter "s" in the word "publications".
Asymmetric Partial Devil's Staircases, a contributed talk in SIAM conference on Applications of Dynamical Systems, Snowbird UT, USA, 2003
Partial devil's staircases (in Hebrew), in The Negev Physics Fete, Sde Boker, Israel, 2003