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Fractal dimension associated with a critical circle map with an arbitrary-order inflection point


Delbourgo, R and Kenny, BG, Fractal dimension associated with a critical circle map with an arbitrary-order inflection point, Physical Review A, 42, (10) pp. 6230-6233. ISSN 1050-2947 (1990) [Refereed Article]

DOI: doi:10.1103/PhysRevA.42.6230


We have studied critical circle maps of the type f, with f =©+-2-z-1/2, -1/2<<1/2, and f(+1)=1+f, where z is the order of the inflection point of the angular variable. Such maps are believed to be useful in the study of the two-frequency quasiperiodic route to chaos. Using a variety of numerical approaches, we have calculated the fractal dimension D associated with such maps as a function of z. The different approaches yield consistent values for D and the completeness of the staircase has also been checked at each order. By comparing 1-D with the width of the mode-locked interval ©(0/1) (which may be analytically determined as a function of z for this class of maps), we have established that the ratio appears to be roughly independent of z with a value of 1/3. This leads to the empirical formula DS1-[(z-1)/3](1/z)z/ (z-1), which predicts for z=3 that DS0.872, in good agreement with previous precise direct numerical determinations. This general result suggests that the average width of mode-locked intervals of cycle length Q declines at the rate of Q-2/D with the 0/1 interval setting the scale for arbitrary intervals for all z and thereby governing the fractal dimension of the set complementary to the devils staircase.

Item Details

Item Type:Refereed Article
Research Division:Physical Sciences
Research Group:Particle and high energy physics
Research Field:Particle physics
Objective Division:Expanding Knowledge
Objective Group:Expanding knowledge
Objective Field:Expanding knowledge in the physical sciences
UTAS Author:Delbourgo, R (Professor Robert Delbourgo)
UTAS Author:Kenny, BG (Mr Brian Kenny)
ID Code:90702
Year Published:1990
Web of Science® Times Cited:7
Deposited By:Mathematics and Physics
Deposited On:2014-04-16
Last Modified:2014-04-16

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