# 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 $$Xn+1=X2n+X0X_{n+1} = X_n^2 + X_0$$.

Where the complex number $$X0X_0$$ is in the Mandelbrot set if $$|Xn|≤2|X_n| \leq 2$$ for all n. Otherwise we assign a color based on $$nn$$ where $$|Xn|>2|X_n| > 2$$.

Now consider another point $$Y0Y_0$$ that gives us $$Yn+1=Y2n+Y0Y_{n+1} = Y_n^2 + Y_0$$.

Let $$Δn=Yn−Xn\Delta_n = Y_n - X_n$$, Then

$$Δn+1=Yn+1−Xn+1=2XnΔn+Δ2n+Δ0\Delta_{n+1} = Y_{n+1} - X_{n+1} = 2X_n \Delta_n + \Delta_n^2 + \Delta_0$$

So far this is crystal clear to me. But now we want to compute $$Δn\Delta_n$$ directly from $$Δ0\Delta_0$$ using pre-computed coefficients of the recursive equation.

The author continues:

Let $$δ=Δ0\delta = \Delta_0$$

$$Δ1=2X0δ+δ2+δ=(2X0+1)δ+δ2Δ2=(4X1X0−2X1−1)δ+((X0−1)2+2X1)δ2+(4X0−2)δ3+o(δ4)\Delta_1 = 2X_0\delta + \delta^2+\delta = (2X_0+1)\delta + \delta^2\\ \Delta_2 = (4X_1X_0 - 2X_1-1)\delta + ((X_0-1)^2+2X_1)\delta^2 + (4X_0-2)\delta^3 + o(\delta^4)$$

Let $$Δn=Anδ+Bnδ2+Cnδ3+o(δ4)\Delta_n=A_n\delta+B_n\delta^2+C_n\delta^3+o(\delta^4)$$

Then

$$An+1=2XnAn+1Bn+1=2XnBn+A2nCn+1=2XnCn+2AnBnA_{n+1} = 2 X_n A_n + 1\\ B_{n+1} = 2 X_n B_n + A_n^2\\ C_{n+1} = 2 X_n C_n + 2 A_n B_n$$

Knowing all $$XnX_n$$ we can pre-compute $$AnA_n$$, $$BnB_n$$, and $$CnC_n$$ and given new point $$Z0Z_0$$ we can compute $$δz\delta_z$$ and searching for $$|Zn|>2|Z_n| > 2$$ is just binary search that is O(log n).

# Question

My question is how to compute the equations for $$AnA_n$$, $$BnB_n$$, and $$CnC_n$$? I tried to “check” the equations by applying the $$Δ\Delta$$ recurrence but I obtained:

$$Δ2=2XnΔ1+Δ21+Δ0==(4X1X0+2X1+1)δ+(2X1+(2X0+1)2)δ2+(4X0+2)δ3+δ4\Delta_2 = 2 X_n \Delta_1 + \Delta_1^2+\Delta_0 =\\ =(4X_1 X_0 + 2 X_1 + 1) \delta + (2X_1 + (2X_0 + 1)^2)\delta^2 + (4X_0+2)\delta^3+\delta^4$$

Which does not match author’s $$Δ2\Delta_2$$.

I have also tried to apply given formulas for $$AnA_n$$, $$BnB_n$$, and $$CnC_n$$ to compute forst few $$Δ\Delta$$‘s but they matched my $$Δ2\Delta_2$$, not authors (for example $$C2=4X0+2).C_2 = 4 X_0 + 2).$$

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

# Bonus

There is probably a general formula for $$Δn\Delta_n$$, can you help me to find it? Something like $$Δn=∑∞i=0C(i)nδi\Delta_n=\sum_{i=0}^\infty C_n^{(i)} \delta^i$$.

# 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!

$\Delta_2 = (4X_1 X_0 + 2 X_1 + 1) \delta + (2X_1 + (2X_0 + 1)^2)\delta^2 + (4X_0+2)\delta^3+\delta^4$
$A_{n+1} = 2 X_n A_n + 1 \\ B_{n+1} = 2 X_n B_n + A_n^2 \\ C_{n+1} = 2 X_n C_n + 2 A_n B_n$
Initialise $A_0 = 1, B_0 = 0, C_0 = 0$ and use their computation for the induction.