Fractals/Iterations in the complex plane/Mandelbrot set/MFpoint12
What is the Myrberg-Feigenbaum point of family ?
Zoom toward Feigenbaum point
The Feigenbaum point (red arrow) is limit of the bifurcation point
Feigenbaum point Julia set
The Myrberg-Feigenbaum point is
- a point c of parameter plane
- a Misiurewicz point
- a biaccesible point. It means that it is a landing point of 2 external rays with irrational angles. The rays are not spiralling at all (no turn), because if the Misiurewicz point is a real number, it does not turn at all
- boundary point between chaotic (-2 < c < MF) and periodic region (MF< c < 1/4)
- the accumulation point is the limit of the disk centers
- it is a limit of a series of bifurcation parameters ( root points ) of period-2n component. In other words period-doubling cascade finishes at the Myrberg-Feigenbaum point.
- it is a limit of a series of band-merging points . In other words a period-doubling cascade of chaotic bands also finishes, from the opposite side, at the MF point.
What is the address of Feigenbaum point ? edit
What is the value of Feigenbaum point ? edit
What external rays land on the Myrberg-Feigenbaum point ? edit
Decimal values of external angles t of rays that lands on the Myrberg-Feigenbaum point are (0.412454... , 0.58755...)
How to compute angles of external rays ? edit
To compute angles one can use 2 methods:
- find a limit of a series of bifurcation parameters ( root points ) of period-2n component.
- find a limit of a series of band-merging points :
How to compute limit of angles landing on bifurcation parameters ? edit
The candidate upper external angle is obtained by using the substitution (string replacing): 0 -> 01 and 1 -> 10 repeatedly:
But it is not known whether the rays actually lands; maybe M is not locally connected at the Feigenbaum point and some long decorations are shielding it from external rays.
One can compute it using Maxima CAS program :
kill(all); remvalue(all); f(x):=if (x=0) then [0,1] else [1,0]; compile(all); a:; a:endcons(,a); for n:2 thru 10 step 1 do ( a:endcons(,a), for x in a[n-1] do ( a[n]:endcons(first(f(x)),a[n]), a[n]:endcons(second(f(x)),a[n])), print(n,a[n]) );
How to compute points of perid n tupling bifurcations ? edit
- UNIVERSALIiTY FOR PERIOD n-TUPLINGS IN COMPLEX MAPPINGS by Predrag CVITANOVIC and Jan MYRHEIM. Physics Letters A Volume 94, Issue 8, 28 March 1983, Pages 329-333
- period tripling:
- point of Golberg - Sinai - Khanin (GSK) lGSK = 0.0236411685 + 0.7836606508i
- Corresponded critical point (GSK point) is situated at λc = 0.0236411685377 + 0.7836606508052i and characterized by following critical indexes , namely, by critical multiplier µc, scale factor α, and parameter scaling constant δ :
- µc = −0.47653179 − 1.05480867i
- α = −2.0969 + 2.3583i
- δ = 4.6002 − 8.9812i
- Near the critical point GSK a structure of "leaves" of the Mandelbrot set possesses a property of scale invariance in respect to rescaling of g-gGSK with the complex factor d=4.60022558 -8.98122473i.
- "sequence of illustrations, each view is centered at the Feigenbaum point and the magnification increases by 4.6692 (the Feigenbaum Constant) each time. The filaments become steadily denser until they fill the view."
- muency: feigenbaum point
- YouTube: Mikhail Lyubich: Story of the Feigenbaum point. Centre International de Rencontres Mathématiques
- fractalforums.org: accumulation-point-of-period-doubling
- On Periodic and Chaotic Regions in the Mandelbrot Set by G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya
- EXTERNAL ARGUMENTS FOR THE CHAOTIC BANDS CALCULATION IN THE MANDELBROT SET by G. Pastor , M. Romera, G. Alvarez, and F. Montoya
- Period tripling accumulation point for complexified Henon map by O.B. Isaeva, S.P. Kuznetsov
- Effect of noise on the period-tripling by Saratov group of theoretical nonlinear dynamics
- Scaling properties in dynamics of non-analytic complex maps near the accumulation point of the period-tripling cascade by O.B. Isaeva, S.P. Kuznetsov