To calculate the sequence rigorously, we can use the following procedure: Divide the Poincare section into a dyadic solenoid(?) according to whether or not the n-th look ahead lands on zero or one. (Some patterns such as 000 are forbidden, so the solenoid has gaps) On each continuous interval compute the curve for n-th look ahead, Fn(x), and see if y=Fn(x) intersects y=x. If yes, and if it isn't a multiple cover of itself, count + 1 and eliminate other similar intervals. It is possible just to search for all intersections of y=Fn(x) with y=x, but using the dyadic solenoid, each interval should have no more than one intersection, making root-solving easier. I will see if I can add the sequence at some point. Maybe it will appear as part of a paper with more similar sequences. Thanks for your interest, --Brad On Wed, Jan 27, 2021 at 11:52 AM Neil Sloane <njasloane@gmail.com> wrote:
Brad, Interesting sequence! Please go ahead and submit it!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jan 27, 2021 at 11:43 AM Brad Klee <bradklee@gmail.com> wrote:
Section 4 of this student paper has a nice tabulation of periodic orbits on the Rössler attractor:
http://chaosbook.org/projects/Carroll/Carr12.pdf
We probably would want to define a sequence:
a(n) counts the number of distinct cycles of winding number n on the Rössler attractor with (a,b,c)=(.2,.2,5.7).
For the numbers a(n), from n=1, I calculated that:
a(n) = 1, 1, 2, 1, 2, 3, 4, 6, 8, 10, 17 . . .
These were determined using a numerical algorithm with randomized searching across a linear Poincare section. If there are hidden cycles, which are difficult to reach by iterating Newton's method, I'm worried these numbers could be wrong.
Does anyone else get the same or better than?
--Brad _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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