c:0.37496784+%i*0.21687214;


graphical analysis


Lets draw 1000 points ( iMax) of critical orbit for this c value using :


It seems that critical orbit tends to period 6 orbit ( aftere 1000 iterations).



Numerical analysis


Finding periodic points and its stability indexes ( using Maxima batch file ):
c=0.21687214*%i+0.37496784
period=1
z=0.43316776849971*%i+0.24966727008436; abs(multiplier(z))=0.99993612079651
z=0.75033272991564-0.43316776849971*%i; abs(multiplier(z))=1.732782180482799
period=2
z=1.065516158748753*%i-0.60176858333837; abs(multiplier(z))=5.567865064661599
z=-1.065516158748753*%i-0.39823141666163; abs(multiplier(z))=5.567865064661598
period=3
z=0.97172269875583*%i+0.44188149039774; abs(multiplier(z))=8.466066508853249
z=0.69783917827279*%i-0.88340148633179; abs(multiplier(z))=12.71325274686825
z=-0.5877486204873*%i-0.64215425816736; abs(multiplier(z))=8.466066508853247
z=0.66838650732078-1.01607219461348*%i; abs(multiplier(z))=12.71325274686825
z=1.075644688759087*%i-0.37401791172118; abs(multiplier(z))=8.466066508853253
z=-1.141385750686922*%i-0.21069434149819; abs(multiplier(z))=12.71325274686825
period=4
z=0.56188415354276*%i+0.61375477558032; abs(multiplier(z))=9.018720343553987
z=-0.30066522761921*%i-0.57374777967294; abs(multiplier(z))=9.018720343553985
z=-0.71951212276901*%i-0.63115487860043; abs(multiplier(z))=23.69960436045009
z=0.87960574273602-0.6834616465094*%i; abs(multiplier(z))=24.60525165965574
z=0.90659030511955*%i+0.43594896254519; abs(multiplier(z))=9.018720343553959
z=0.51355457194445*%i-0.87657070513511; abs(multiplier(z))=24.60525165965569
z=-1.126446045227947*%i-0.13168958833018; abs(multiplier(z))=24.60525165965567
z=0.42913986446643-1.091001256587128*%i; abs(multiplier(z))=23.69960436044991
z=1.007326345940794*%i-0.25688664339254; abs(multiplier(z))=9.018720343554017
z=0.7920930475885*%i-0.82558065657116; abs(multiplier(z))=23.69960436045016
z=0.68155428040484-0.98548143841898*%i; abs(multiplier(z))=24.60525165965581
z=1.12511931299562*%i+0.25562662596955; abs(multiplier(z))=23.69960436045015
period=5
z=0.30392692109477*%i+0.54238969322179; abs(multiplier(z))=5.987981218776015
z=-0.1031518041288*%i-0.42197410811189; abs(multiplier(z))=5.987981218776088
z=-0.38459580975331*%i-0.6340832352332; abs(multiplier(z))=31.06753529820118
z=0.54656579898887*%i+0.57678284594709; abs(multiplier(z))=5.987981218775902
z=1.103426228726584*%i+0.27428778789849; abs(multiplier(z))=31.06753529822347
z=0.4387290318116*%i-0.83105471131174; abs(multiplier(z))=44.18802941343628
z=-0.63380306829549*%i-0.65660305743832; abs(multiplier(z))=49.2677239068064
z=0.8731366098391-0.51234351775255*%i; abs(multiplier(z))=44.18802941343557
z=0.90987324959461*%i-0.1758618270574; abs(multiplier(z))=5.987981218775849
z=0.56736465952901*%i-0.88190595808821; abs(multiplier(z))=49.92539527908808
z=-0.76194005830615*%i-0.60035874181269; abs(multiplier(z))=49.92539527908504
z=0.70460365061102*%i+0.629115452324; abs(multiplier(z))=31.0675352982014
z=0.84737169407629*%i+0.40891211875447; abs(multiplier(z))=5.987981218774501
z=-1.102787705126374*%i-0.10058912099819; abs(multiplier(z))=44.18802941475932
z=1.049186204913203*%i+0.40438908565656; abs(multiplier(z))=49.26772390677004
z=0.45081060277147-1.085613550661432*%i; abs(multiplier(z))=49.92539527446423
z=0.28780262782769-1.044931302917478*%i; abs(multiplier(z))=31.06753529776084
z=1.131746889482733*%i+0.1548458064193; abs(multiplier(z))=49.92539527908263
z=0.82218481877284*%i-0.76734781165161; abs(multiplier(z))=31.06753529808856
z=-1.085088751563241*%i-0.37031878437524; abs(multiplier(z))=72.54693760543108
z=0.87483949925823-0.67781962432688*%i; abs(multiplier(z))=44.18802941335857
z=1.088267512116833*%i-0.39083920029948; abs(multiplier(z))=49.26772390670064
z=0.7277771601398*%i-0.87696099797625; abs(multiplier(z))=72.54693760646659
z=-0.98129737358586*%i-0.44400168367555; abs(multiplier(z))=49.26772390690442
z=0.83082330202931-0.7838524072947*%i; abs(multiplier(z))=49.92539527955116
z=1.065431040176901*%i-0.5622933199819; abs(multiplier(z))=49.26772390717106
z=1.020529634837456*%i-0.66531375670817; abs(multiplier(z))=72.54693760727088
z=0.61436883714881-1.059592229321916*%i; abs(multiplier(z))=72.54693760712381
z=0.6808725463406-0.96909462146839*%i; abs(multiplier(z))=44.1880294154607
z=-1.141072670369937*%i-0.22387050071707; abs(multiplier(z))=72.54693760401455

Find z : abs(multiplier(z))< 1 and you will find a period.

Lets check it numerically ( iMax=1000) : In Maxima : GivePeriodOfC(c,eps);



graphical analysis using c program



Lets draw more point. Here is 2D image of 100 000 points :


So my previous results are bad. (:-|

root point of period 6 component on internal ray 1/6 is :
c_r= 0.375 + 0.21650635094611*i

So point c = 0.37496784+0.21687214*i is above root point :


c-c_r = 3.6578905388998106*10^-4*%i-3.2160000000003297*10^-5
abs(c-c_r)=3.67200078357464*10^-4


Here is 3D image of 100 000 points :


It is possible to color 6 (pseudo)spirals of critical orbit (proposition of Wolf Jung):



Compare with 3d image in Maple by Filip Piekniewski
Lets find fixed points using Maxima :
(%i6) fixed:solve([z=f(z,c)], [z]);
`rat' replaced -0.37496784 by -2915/7774 = -0.374967841523
`rat' replaced -0.21687214 by -2797/12897 = -0.2168721408079
(%o6) [z=-(sqrt(32959)*sqrt(-43487756*%i-25058871)-1285401)/2570802,z=(sqrt(32959)*sqrt(-43487756*%i-25058871)+1285401)/2570802]
(%i4) float(fixed), numer;
(%o4) [z=-3.8898367124344855*10^-7*(181.5461373866159*(-4.3487756*10^+7*%i-2.5058871*10^+7)^0.5-1285401.0),z=3.8898367124344855*10^-7*(181.5461373866159*(-4.3487756*10^+7*%i-2.5058871*10^+7)^0.5+1285401.0)]
(%i5) rectform(fixed);
(%o5) [z=0.4331677702216*%i+0.24966727014686,z=0.75033272985314-0.4331677702216*%i]
(%i7) cabs(2*fixed[1]);
(%o7) 2*abs(z)=sqrt(sqrt(2519131937710177)/100261278+(1-(sqrt(32959)*sqrt(sqrt(2519131937710177)-25058871))/(1285401*sqrt(2)))^2+1943/7774)
(%i8) float(%), numer;
(%o8) 2.0*abs(z)=0.99993612384259
(%i9) cabs(2*fixed[2]);
(%o9) 2*abs(z)=sqrt(sqrt(2519131937710177)/100261278+(sqrt(sqrt(2519131937710177)-25058871)/(39*sqrt(2)*sqrt(32959))+1)^2+1943/7774)
(%i10) float(%), numer;
(%o10) 2.0*abs(z)=1.732782182096313

It shows that attracting fixed point is :
z=0.4331677702216*%i+0.24966727014686
with modulus of multiplier = 0.99993612384259
Because modulus of multiplier is smaller and near equal 1 so this point is weakly attracting.
Here (internal angle > 1/6 ) forward orbit of critical point is turning around fixed point in the counterclockwise direction instead of going straight to it.
If the point will be below internal ray 1/6 ( internal angle < 1/6 ) then orbit will be turning around fixed point in the clockwise direction
See what happens when one goes along internal ray 1/6


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Feel free to e-mail me!

Author: Adam Majewski
adammaj1-at-o2-one_dot-pl

http://fraktal.republika.pl

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