Fractals/Iterations in the complex plane/construct
How to construct a map with desired properities ?
Shape
[edit | edit source]Shape of
- Julia set
- critical orbit
- external rays landind on the repelling cycle ( spirals)
Relation between shape types and dynamics:
- n-th arm spiral: attracting or repelling n-periodic orbit ( cycle)
- closed curve: Siegel disc ( rotation)
- n-th arm star = period n parabolic root
Modelling or shaping
[edit | edit source]Usually one should controll 2 parameters:
- fixed point
- period p orbit
-
Julia set is modelled by critical orbit
-
Julia set is modelled by critical orbit
-
Siegel disk critical orbit is modelled by period 7 repelling orbit
-
Siegel disk critical orbit is modelled by period 2 repelling orbit
-
Julia set is modelled by repelling period 6 orbit and it's external rays
See
Conversions and evolutions
[edit | edit source]-
from Siegel to parabolic star
-
Superattracting through attracting spiral to parabolic star ( along internal ray)
Examples
[edit | edit source]- Rational functions with prescribed critical points by I. Scherbak
- constructing map with Fatou components of desired type[1]
- Constructing polynomials whose Julia set resemble a desired shape[2]
- Constructing polynomials that have an attracting cycle visiting predefined points in the plane [3]
- constructuing QUARTIC JULIA SETS INCLUDING ANY TWO COPIES OF QUADRATIC JULIA SETS
- constructing critically finite real polynomial maps with specified combinatorics[4]
roots
[edit | edit source]The fundamental theorem of algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots[5][6]
The factor theorem[7] states that a polynomial has a factor if and only if (i.e. is a root of multiplicity[8] m)
Examples
The polynomial[9]
The polynomial
has roots:
- 1 of multiplicity 2
- -4 of multiplicity 1
rboyce1000
[edit | edit source]p(z) = z^4 + O(z^2), where the four roots of p(z) are:
- one fixed at the origin,
- the remaining three forming the vertices of an equilateral triangle centered at the origin and rotating.
create polynomial with desired properities
- f(z) = z*g(z) with root at origin
- g(z) is a 3-rd root of unity =
f(z) = z(z^2+z+1)
One can check it with Maxima CAS
(%i1) solve([z*(z^2+z+1)=0],[z]); sqrt(3) %i + 1 sqrt(3) %i - 1 (%o1) [z = - --------------, z = --------------, z = 0] 2 2 (%i2)
to rotate it around origin let's change 1 with : wher t is a proper fraction in turns
See : Vector_field short videos by boyce1000
one parameter
[edit | edit source]System of 2 equations:
where:
- is a rational function with one parameter c
- is the -fold composition of with itself
- is a cyclic point ( point of limit cycle)
- p is a period of the cycle
- is a multiplier[10] ( complex number)
- is a stability of the cycle ( Real number )
Input :
- function
- p ( integer)
- r ( real number)
- ( real number or rational number)
Unknowns ( solutions or output):
- parameter c ( complex number)
- periodic point ( complex number)
Maxima CAS program:
/* batch("m.mac"); */ display2d:false$ kill(all)$ ratprint:false$ /* complex quadratic polynomial */ f(z,c):= z*z+c $ /* iterated function */ F(z, c, n) := if n=1 then f(z,c) else f(F(z, c, n-1),c)$ /* multiplier = first deric=vative */ m(z,c,p):= diff(F(z,c,p),z,1)$ l(r,t) := float(rectform(r*exp(2*%pi*t*%i)))$ /* input */ p:5$ r:1.0$ t:0$ /* system of equations */ e1: F(z,c,p)=z; e2: m(z,c,p)=l(r,t); /* output = solutions = 2 complex number: c, z */ s:solve([e1,e2])$ s:map('float,s)$ s:map('rectform,s);
Example output:
For :
- p = 3
- r=1.0
- t=0.0
[ [z = 0.5,c = 0.25], [z = (-0.4330127018922193*%i)-0.25,c = (-0.6495190528383289*%i)-0.125], [z = 0.4330127018922193*%i-0.25,c = 0.6495190528383289*%i-0.125], [z = -0.05495813133539004,c = -1.75], [z = 1.301937809824245,c = -1.75], [z = -1.746979634104245,c = -1.75] ]
For :
- p = 5
- r=1.0
- t=0.0
[ [z = 0.5,c = 0.25], [z = 0.4755282581475767*%i+0.1545084971874737,c = 0.3285819450744551*%i+0.3567627457812099], [z = 0.1545084971874737-0.4755282581475767*%i, c = 0.3567627457812106-0.3285819450744586*%i], [z = 0.2938926261462365*%i-0.4045084971874737, c = 0.5316567552200239*%i-0.4817627457812153], [z = (-0.2938926261462365*%i)-0.4045084971874737, c = (-0.5316567552199369*%i)-0.481762745781224], [z = -0.003102011282477321,c = -1.985409652076318], [z = 0.0109289978340113,c = -1.860587002096436], [z = 8.008393221517376E-4*%i-0.01213161194929343, c = 1.100298437397382*%i-0.1978729466687337], [z = (-8.008393221517376E-4*%i)-0.01213161194929343, c = (-1.100298437397305*%i)-0.1978729466687667], [z = 0.02151217276434695*%i-0.005267866463337371, c = 0.3797412022535638*%i-1.256801993945385], [z = (-0.02151217276434695*%i)-0.005267866463337371, c = (-0.3797412022517599*%i)-1.256801993944077], [z = 0.02591758988716001*%i+0.0096648625988135, c = 0.9868115621249533*%i-0.04506136597934137], [z = 0.0096648625988135-0.02591758988716001*%i, c = (-0.9868115621250132*%i)-0.04506136597930513], [z = -0.02506558296814108,c = -1.624396967608546], [z = 0.02532354987824971*%i-0.0286751769590709, c = 0.6415066667139064*%i+0.3599331333357185], [z = (-0.02532354987824971*%i)-0.0286751769590709, c = 0.3599331333357186-0.6415066667139071*%i], [z = 0.7018214526647177,c = -1.860587002096436], [z = 0.5745382937725365*%i+0.1798116252110209, c = (-0.379741202251533*%i)-1.25680199394442], [z = 0.1798116252110209-0.5745382937725365*%i, c = 0.3797412022514344*%i-1.256801993944486], [z = -0.5997918293000261,c = -1.624396967608546], [z = 0.6400543521659254*%i+0.3601141169309163, c = 0.6415066667138928*%i+0.3599331333356947], [z = 0.3601141169309163-0.6400543521659254*%i, c = 0.3599331333356951-0.6415066667138929*%i], [z = 0.747361547631752*%i+0.4122389750905872, c = 0.3599331333377524-0.6415066667118048*%i], [z = 0.4122389750905872-0.747361547631752*%i,c = 0.6415066667118131*%i+0.3599331333377574], [z = -1.264646754738656,c = -1.624396967608546], [z = 0.838427461519175*%i+0.1867295812979602,c = (-0.9868115621248*%i)-0.04506136597962632], [z = 0.1867295812979602-0.838427461519175*%i, c = 0.9868115621248269*%i-0.04506136597961512], [z = 1.012227741688957,c = -1.624396967608546], [z = 0.6736931444481549*%i-0.7131540376767388, c = 0.9868115621009495*%i-0.04506136566593825], [z = (-0.6736931444481549*%i)-0.7131540376767388, c = (-0.9868115621015654*%i)-0.04506136566602404], [z = 0.6816651712455555*%i+0.8064792250322852, c = (-1.100298438532418*%i)-0.1978729463920518], [z = 0.8064792250322852-0.6816651712455555*%i,c = 1.100298438531886*%i-0.197872946387467], [z = 0.9873125420152975*%i-0.04563967787575593, c = 0.9868115621249436*%i-0.04506136597927069], [z = (-0.9873125420152975*%i)-0.04563967787575593, c = (-0.9868115621249249*%i)-0.04506136597929692], [z = -1.368033648790746,c = -1.860587002096436], [z = -1.623768668573244,c = -1.624396967608546], [z = 1.600752508361204,c = -1.860587002096436], [z = 0.8177857184842046*%i-0.8491638964763748, c = 0.6415066726649287*%i+0.3599331357137042], [z = (-0.8177857184842046*%i)-0.8491638964763748, c = 0.3599331357115682-0.6415066726792946*%i], [z = -1.860467532467532,c = -1.860586580956207], [z = 0.1585230889211015*%i+1.129895436404861, c = (-0.3797412017812437*%i)-1.256801993890818], [z = 1.129895436404861-0.1585230889211015*%i, c = 0.3797412020742688*%i-1.256801993924219], [z = 1.102491882350288*%i+0.07994573682221373, c = 0.641506666713125*%i+0.3599331333375105], [z = 0.07994573682221373-1.102491882350288*%i, c = 0.3599331333375118-0.641506666713142*%i], [z = 1.10027900645412*%i-0.1977264120044163,c = 1.100298437399976*%i-0.1978729466589521], [z = (-1.10027900645412*%i)-0.1977264120044163, c = (-1.100298437392994*%i)-0.1978729466579122], [z = 0.3795145554958574*%i-1.257237017109811, c = 0.3797412012322979*%i-1.256801993538778], [z = (-0.3795145554958574*%i)-1.257237017109811, c = (-0.3797412011893692*%i)-1.256801993401957], [z = 0.8966903093631682*%i-1.01776444141452, c = 0.986811439368143*%i-0.04506141337632084], [z = (-0.8966903093631682*%i)-1.01776444141452, c = (-0.9868114393633113*%i)-0.04506141338736716], [z = 1.407944514501891,c = -1.985409652076318], [z = 0.7215120925377011*%i+1.234881318742427, c = (-1.100298500720014*%i)-0.1978727350763138], [z = 1.234881318742427-0.7215120925377011*%i,c = 1.100298500782114*%i-0.1978727352231734], [z = 0.6651899971189704*%i-1.369391104706556, c = 1.100298438532065*%i-0.1978727774731155], [z = (-0.6651899971189704*%i)-1.369391104706556, c = (-1.100298478086625*%i)-0.1978727911942495], [z = 0.1731238730127708*%i-1.554564024233688, c = 0.3797412149717089*%i-1.256801976456581], [z = (-0.1731238730127708*%i)-1.554564024233688, c = (-0.3797411926534995*%i)-1.256801968631482], [z = 1.842105908761944,c = -1.985410334346504], [z = 1.956403762662807,c = -1.985409652076318] ]
Mandelbrot Set - Convergent Evolution of P/Q Limbs in Seahorse Valley
[edit | edit source]Mandelbrot Set - Convergent Evolution of P/Q Limbs in Seahorse Valley by izaytsev0
- main cardioid seahorse valley = Gap between the head ( period 2 component) and the body (or shoulder = main cardioid). Particularly the upper one part
- 2 windows
- left: limbs from period 2 component
- right: limbs from period 1 componnet
- in each window one can see limb p/q from 14/30 on the right and increasing p ?
Compare with Real dense fractal Zoom! Part 2 by SeryZone Arts
See also
[edit | edit source]References
[edit | edit source]- ↑ fractalforums: julia-sets-true-shape-and-escape-time
- ↑ fractalforums : constructing-polynomials-whose-julia-set-resemble-a-desired-shape
- ↑ fractalforums : constructing-polynomials-with-attracting-cycles
- ↑ The W. Thurston algorithm applied to real polynomial maps Araceli Bonifant, J. Milnor, S. Sutherland Published 15 May 2020
- ↑ The fundamental theorem of algebra in wikipedia
- ↑ Ed Pegg Jr "The Fundamental Theorem of Algebra" http://demonstrations.wolfram.com/TheFundamentalTheoremOfAlgebra/ Wolfram Demonstrations Project Published: November 10 2011
- ↑ Factor theorem in wikipedia
- ↑ Multiplicity_of_a_root_of_a_polynomial in wikipedia
- ↑ fractalforums.org : julia-sets-true-shape-and-escape-time
- ↑ Stability_of_periodic_points_(orbit)_-_multiplier in wikipedia
- Bishop, Christopher J. Constructing entire functions by quasiconformal folding. Acta Math. 214 (2015), no. 1, 1--60. doi:10.1007/s11511-015-0122-0.
- Constructing entire functions with non-locally connected Julia set by quasiconformal surgery by Yanhua Zhang, Gaofei Zhang
- Kumar, Rajen & Nayak, Tarakanta. (2020). The real non-attractive fixed point conjecture and beyond.
- Specifying attracting cycles for Newton maps of polynomials by James T. Campbell, Jared T. Collins
- Exploring the Mandelbrot set. The Orsay Notes. Adrien Douady John H. Hubbard: constructing a polynomial with a given tree
- Godillon, S'ebastien. “Construction of rational maps with prescribed dynamics.” (2010).
- On building polynomials, November 2005The Mathematical Gazette 89(516), DOI: 10.2307/3621936, Christopher Sangwin