10 Circle Maps: Irrationally Winding - Center for Nonlinear Science

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4 P. Cvitanovic winding number is irrational. A plot of the winding number W as a function of the shift parameter is a convenient visualization of the mode-locking structure of circle maps. It yields a monotonic \devil's staircase" of g. 10.1 whose selfsimilar structure we are to unravel. Fig. 10.1. The critical circle map (k = 1 in (1)) devil's staircase[5] the winding number W as function of the parameter . Circle maps with zero slope at the inection point x c f 0 (x c )=0f 00 (x c )=0 (k = 1, x c = 0 in (1)) are called critical: they delineate the borderline of chaos in this scenario. As the non-linearity parameter k increases, the mode-locked intervals become wider, and for the critical circle maps (k =1)they ll out the whole interval[16]. A critical map has a superstable P=Q cycle for any rational P=Q, as the stability ofany cycle that includes the inection point equals zero. If the map is non-invertable (k > 1), it is called supercritical the bifurcation structure of this regime is extremely rich and beyond the scope of these (and most other such) lectures. For physicists the interesting case is the critical case the shift map is \easy" number theory (Farey rationals, continued fractions) which one uses as a guide to organization of the non-trivial critical case. In particular, the problem of organizing subcritical mode lockings reduces to the problem of organizing rationals on the unit interval. The self-similar structure of the devil's staircase suggests a systematic way of separating the mode lockings into hierarchies of levels. The set of rationals P=Q clearly possesses rich number-theoretic structure, which we shall utilize here to formulate three dierent partitionings of rationals: 1. Farey series 2. Continued fractions of xed length 3. Farey tree levels
10. Circle Maps 5 10.2 Farey Series Partitioning Intuitively, the longer the cycle, the ner the tuning of the parameter required to attain it given nite time and resolution, we expect to be able to resolve cycles up to some maximal length Q. This is the physical motivation for partitioning[19] mode lockings into sets of cycle length up to Q. In number theory such set of rationals is called a Farey series. (10.1) Denition. The Farey series[21] F Q of order Q is the monotonically increasing sequence of all irreducible rationals between 0 and 1 whose denominators do not exceed Q. Thus P i =Q i belongs to F Q if 0
Page 1 and 2: 10 Circle Maps: Irrationally Windin
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Page 9 and 10: 10. Circle Maps 9 (11) f a (x) = f
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Page 17 and 18: 10. Circle Maps 17 that the leading
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10 Circle Maps: Irrationally Winding - Center for Nonlinear Science

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