Perturbation of Mandelbrot set fractal

I recently discovered very clever technique how co compute deep zooms of the Mandelbrot set using Perturbation and I understand the idea very well but when I try to do the math by myself I never got the right answer.

I am referring to original PDF by K.I. Martin but I will put the necessary equations below.

Theory

Mandelbrot set is defined as Xn+1=X2n+X0.

Where the complex number X0 is in the Mandelbrot set if |Xn|2 for all n. Otherwise we assign a color based on n where |Xn|>2.

Now consider another point Y0 that gives us Yn+1=Y2n+Y0.

Let Δn=YnXn, Then

Δn+1=Yn+1Xn+1=2XnΔn+Δ2n+Δ0

So far this is crystal clear to me. But now we want to compute Δn directly from Δ0 using pre-computed coefficients of the recursive equation.

The author continues:

Let δ=Δ0

Δ1=2X0δ+δ2+δ=(2X0+1)δ+δ2Δ2=(4X1X02X11)δ+((X01)2+2X1)δ2+(4X02)δ3+o(δ4)

Let Δn=Anδ+Bnδ2+Cnδ3+o(δ4)

Then

An+1=2XnAn+1Bn+1=2XnBn+A2nCn+1=2XnCn+2AnBn

Knowing all Xn we can pre-compute An, Bn, and Cn and given new point Z0 we can compute δz and searching for |Zn|>2 is just binary search that is O(log n).

Question

My question is how to compute the equations for An, Bn, and Cn? I tried to “check” the equations by applying the Δ recurrence but I obtained:

Δ2=2XnΔ1+Δ21+Δ0==(4X1X0+2X1+1)δ+(2X1+(2X0+1)2)δ2+(4X0+2)δ3+δ4

Which does not match author’s Δ2.

I have also tried to apply given formulas for An, Bn, and Cn to compute forst few Δ‘s but they matched my Δ2, not authors (for example C2=4X0+2).

What am I doing wrong? Is it something with complex numbers?

Bonus

There is probably a general formula for Δn, can you help me to find it? Something like Δn=i=0C(i)nδi.

Edit

Anybody? The “Tumbleweed” badge for this question is cool but I thought this should be rather “simple” problem. The solution should probably involve Taylor series, I just need to point out to the right direction. Thanks!

Answer

Your computation is correct:

Δ2=(4X1X0+2X1+1)δ+(2X1+(2X0+1)2)δ2+(4X0+2)δ3+δ4

Their computation for the induction is correct:

An+1=2XnAn+1Bn+1=2XnBn+A2nCn+1=2XnCn+2AnBn

Initialise A0=1,B0=0,C0=0 and use their computation for the induction.

Attribution
Source : Link , Question Author : NightElfik , Answer Author : Neal Lawton

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