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C00002 00002	rotati[e83,jmc]		Rotation number problem for Ping Wong on LOTS
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rotati[e83,jmc]		Rotation number problem for Ping Wong on LOTS

Ping Wong
Dept. of Aero and Astro
7-4673
P.PING@LOTSA

Consider T(x) = x + λ + αx(1-x)
with the last term extended beyond  0 ≤ x ≤ 1 by making it periodic.
Let  rot(λ)  be the function giving the rotation number of the transformation
as a function of λ.  Let  I(p/q)  be the length of the interval of  λ
for which the rotation number is p/q.  We are interested in how
I(p/q) → 0  as  q  gets large, and we want experimental evidence for
interesting values of  p/q  and α.  I suspect that there may be
some "universal behavior" a la Mitchell Feigenbaum.  To verify this would
require considering some other functions besides  x(1-x), though
maybe varying  α  would give somm indication.

Wong should write a program to compute I(p/q) for input values of p/q
and values of the parameter α.  It should be convenient to replace
x(1-x) by some other function.  (sin(x/π))↑2 would be smooth, and this
might make some difference.

Here is some program and some data values.  The numerical output is the
result of executing the LISP DO statement.  The fact that the result
varies below and above 1.0 shows that there are periodic solutions.
Actually there are probably two periodic solutions, and looking at the
transformation of the domain with finer resolution would show that some
of them are just the iterates of the others.  It looks like the rotation
number is 3/4 in this case where the parameters are  lam=.13, alpha =.8,
and  q=4, where  q  is the number of iterations.

(defun f1 (x) (plus x lam (times alpha (red x) (difference 1.0 (red x)))))
(defun red (x) (difference x (fix x)))
(defun iter (x) (do ((i 0 (add1 i))
		    (r x (f1 r)))
		   ((= i q) r)))
(setq lam 0.13)
(setq alpha 0.8)
(setq q 4)
(setq deltax 0.01)

(do ((x 0.0 (plus x deltax))
     (lim (difference 1.0 (times .5 deltax)))
     )
    ((greaterp x lim))
    (print (list x (iter x) (difference (iter x) x))))

(setq a (quotient (difference y1 y0) deltax))
(setq b (difference y0 (times a x0)))
(setq xx (quotient (difference y b) a))
(0.0 0.97144531 0.97144531) 
(0.01 0.99594653 0.98594653) 
(0.02 1.01719604 0.99719604) 
(0.03 1.03559437 1.00559437) 
(0.04 1.05149628 1.01149628) 
(0.0500000003 1.06521554 1.01521555) 
(0.0600000005 1.07702944 1.01702943) 
(0.07 1.08718255 1.01718256) 
(0.08 1.09589045 1.01589045) 
(0.09 1.10334277 1.01334277) 
(0.1 1.10970621 1.00970621) 
(0.109999999 1.11512695 1.00512695) 
(0.119999999 1.11973312 0.99973313) 
(0.129999999 1.12363677 0.99363677) 
(0.139999999 1.12693574 0.986935735) 
(0.149999999 1.12971534 0.97971534) 
(0.159999998 1.14918272 0.989182726) 
(0.169999998 1.16911684 0.999116845) 
(0.179999998 1.18739763 1.00739764) 
(0.189999998 1.20415726 1.01415727) 
(0.199999997 1.21951692 1.01951692) 
(0.209999997 1.23358782 1.02358782) 
(0.219999997 1.24647206 1.02647206) 
(0.229999997 1.25826357 1.02826358) 
(0.239999996 1.2690487 1.02904871) 
(0.249999996 1.27890699 1.02890699) 
(0.259999998 1.28791161 1.02791162) 
(0.27 1.29613018 1.02613018) 
(0.28 1.30362496 1.02362496) 
(0.290000003 1.31045349 1.02045348) 
(0.300000004 1.31666891 1.01666892) 
(0.310000006 1.32232028 1.01232028) 
(0.320000008 1.32745305 1.00745304) 
(0.33000001 1.3321091 1.00210908) 
(0.34000001 1.33632734 0.99632733) 
(0.350000013 1.3401437 0.99014369) 
(0.360000014 1.34359135 0.98359133) 
(0.370000016 1.34670112 0.9767011) 
(0.380000018 1.34950148 0.96950146) 
(0.39000002 1.36452055 0.974520534) 
(0.40000002 1.38590218 0.98590216) 
(0.410000023 1.40597844 0.99597842) 
(0.420000024 1.42483082 1.0048308) 
(0.430000026 1.44253501 1.01253499) 
(0.440000027 1.45916156 1.01916154) 
(0.45000003 1.47477613 1.0247761) 
(0.46000003 1.48943983 1.0294398) 
(0.470000032 1.50320965 1.03320962) 
(0.480000034 1.51613882 1.03613879) 
(0.490000036 1.52827689 1.03827685) 
(0.50000004 1.53967032 1.03967029) 
(0.510000035 1.55036226 1.04036222) 
(0.52000003 1.56039329 1.04039326) 
(0.53000003 1.56980133 1.0398013) 
(0.54000003 1.5786218 1.03862178) 
(0.55000003 1.58688806 1.03688803) 
(0.560000025 1.59463127 1.03463125) 
(0.57000002 1.60188076 1.03188074) 
(0.58000002 1.608664 1.028664) 
(0.59000002 1.61500694 1.02500692) 
(0.60000002 1.62093392 1.02093391) 
(0.610000014 1.62646794 1.01646793) 
(0.62000001 1.6316306 1.0116306) 
(0.63000001 1.6364424 1.00644238) 
(0.64000001 1.64092258 1.00092258) 
(0.650000006 1.64508946 0.99508946) 
(0.660000004 1.64896034 0.98896033) 
(0.67 1.65255158 0.982551575) 
(0.68 1.65587872 0.97587872) 
(0.69 1.65895656 0.96895656) 
(0.699999996 1.66179906 0.96179906) 
(0.70999999 1.67910548 0.96910548) 
(0.71999999 1.70123567 0.981235676) 
(0.72999999 1.72193986 0.99193987) 
(0.73999999 1.74131288 1.0013129) 
(0.749999985 1.75944284 1.00944285) 
(0.75999998 1.77641138 1.01641141) 
(0.76999998 1.79229452 1.02229454) 
(0.77999998 1.8071627 1.02716273) 
(0.78999998 1.82108125 1.03108127) 
(0.799999975 1.834111 1.03411104) 
(0.80999997 1.84630834 1.03630836) 
(0.81999997 1.85772555 1.03772558) 
(0.82999997 1.86841123 1.03841126) 
(0.83999997 1.87841046 1.0384105) 
(0.849999964 1.88776511 1.03776515) 
(0.85999996 1.89651392 1.03651397) 
(0.86999996 1.90469293 1.03469297) 
(0.87999996 1.91233552 1.03233556) 
(0.889999956 1.91947249 1.02947253) 
(0.899999954 1.92613262 1.02613267) 
(0.90999995 1.93234234 1.02234238) 
(0.91999995 1.93812618 1.01812623) 
(0.92999995 1.94350682 1.01350687) 
(0.939999945 1.94850518 1.00850524) 
(0.94999994 1.95314051 1.00314057) 
(0.95999994 1.95743062 0.997430675) 
(0.96999994 1.96139178 0.99139184) 
(0.97999994 1.96503906 0.98503912) 
(0.989999935 1.96838604 0.978386104) 
NIL 
0.83983317 
0.98039938 
0.0233387095 
(setq xx (f1 xx))
0.171573922 
0.415282972 
0.7395414 
1.02363734