Fractals/island wake

The biggest island of the p/q wake

• for p/q <= 1/3
  • is located on the first branch ( reading from the left to right)
  • has period = q+1
  • external rays ?
• for p/q > 1/3
  • is located on the last branch ( reading from the left to right)
  • has period = q+2

Examples

Wake 1/2

period 3 island is the biggest island of wake 1/2

• 
  Period 3 island
• 
  Period 3 island is the biggest island of wake 1/2

find angles of some child bulbs of period 3 component ( island) on main antenna with external rays (3/7,4/7)

Plane description :

-1.76733 +0.00002 i @ 0.05

One can check it using program Mandel by Wolf Jung :

The angle  3/7  or  p011 has  preperiod = 0  and  period = 3.
The conjugate angle is  4/7  or  p100 .
The kneading sequence is  AB*  and the internal address is  1-2-3 .
The corresponding parameter rays are landing at the root of a primitive component of period 3.

See also:

• $F_{1/2}$ family: real slice of Mandelbrot set.
  • periodic part: period doubling cascade. Escape route 1/2
  • the Myrberg-Feigenbaum point of $F_{1/2}$ family
  • chaotic part main antenna is a shrub of $F_{1/2}$ family

1/3

Period 4 island

• Center X -0.15710375803
• Center Y +1.03258348530
• Pixel step +0.0000375

wake 1/7

• 

12/25

wake 12/25

• root c = -0.738203140939397 +0.124839088573366 i
• The 12/25-wake of the main cardioid is bounded by the parameter rays with the angles
  • 11184809/33554431 or p0101010101010101010101001 and
  • 11184810/33554431 or p0101010101010101010101010 .
• the center of the satellite component c = -0.739829393511579 +0.125072144080321 i period = 25

biggest island of 12/25 wake

• cardioid
  • center c = -0.744245042107463 +0.127908444364520 i
  • period = 27
  • cusp

Size

• miniset-and-embedded-julia-size-estimates
• Windows of periodicity scaling by E Demidov
• J.A.Yorke, C.Grebogi, E.Ott, and L.Tedeschini-Lalli "Scaling Behavior of Windows in Dissipative Dynamical Systems" Phys.Rev.Lett. 54, 1095 (1985)
• B.R.Hunt, E.Ott Structure in the Parameter Dependence of Order and Chaos for the Quadratic Map J.Phys.A 30 (1997), 7067.

Haskell program

size formula degree p a b = do
  -- z = x + i y = 0
  x <- real
  y <- real
  x .= 0
  y .= 0
  -- matrix L
  lxa <- real
  lxb <- real
  lya <- real
  lyb <- real
  -- L = identity
  lxa .= 1
  lxb .= 0
  lya .= 0
  lyb .= 1
  -- matrix B
  bxa <- real
  bxb <- real
  bya <- real
  byb <- real
  -- B = identity
  bxa .= 1
  bxb .= 0
  bya .= 0
  byb .= 1
  -- loop one period
  j <- int
  j .= 1
  while_ (j .< p) $ do
    -- allocate next z, l
    xn <- real
    yn <- real
    lxan <- real
    lxbn <- real
    lyan <- real
    lybn <- real
    -- calculate next z, l
    let R2 fx fy = formula (R2 a b) (R2 x y)
        -- calculate derivatives
        fdxa = ddx d x fx
        fdxb = ddx d y fx
        fdya = ddx d x fy
        fdyb = ddx d y fy
        d v u
              | u == x && v == x = lxa
              | u == x && v == y = lxb
              | u == y && v == x = lya
              | u == y && v == y = lyb
              | u == v = 1
              | otherwise = 0
    -- z = f(z, c)
    xn .= fx
    yn .= fy
    x .= xn
    y .= yn
    -- L = J_f(z, c)
    lxan .= fdxa
    lxbn .= fdxb
    lyan .= fdya
    lybn .= fdyb
    lxa .= lxan
    lxb .= lxbn
    lya .= lyan
    lyb .= lybn
    -- B = B + 1/L
    det <- real
    det .= lxa * lyb - lxb * lya
    bxa .= bxa + lyb / det
    bxb .= bxb - lxb / det
    bya .= bya - lya / det
    byb .= byb + lxa / det
    -- loop counter
    j .= j + 1
  -- l = sqrt (abs (det L))
  l <- real
  l .= sqrt (abs (lxa * lyb - lxb * lya))
  -- beta = sqrt (abs (det B))
  beta <- real
  beta .= sqrt (abs (bxa * byb - bxb * bya))
  -- compute l^d b
  d <- float
  d .= degree / (degree - 1)
  llb <- real
  llb .= l ** d * beta
  return_ (1 / llb)