$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i²-0.5
Center:-1.0314-0.0030i; Zoom = 32

Today's picture is the first of several zooms into yesterday's, Sequence Fractals Part III #28.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i²-0.5
Center:-1.00141+0.00739i; Zoom = 320

This image is a zoom into the upper right of yesterday's, Sequence Fractals Part III #29 .

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i²-0.5
Center:-1.00207+0.00630i; Zoom = 2048

Another step deeper, a little left of center and near the bottom of yesterday's picture, Sequence Fractals Part III #30

$ z^2+c+s_i$
s₀=-0.37, s_i+1=s_i²-0.5
Center:0.216+1.032i; Zoom = 32

The same sequence formula as the last few is used today, but with a different start point. The start point is very close to the fixed point for the sequence. The perturbation from the standard formula is small, and indeed the big picture view (not shown) is very close to the Mandelbrot set.

This image is taken off of the main branch point above the period three bulb. Without the perturbation, you would find a large mini at this location.

$ z^2+c+s_i$
s₀=0.37, s_i+1=-0.5*s_i²+1.7
Center:-0+0i; Zoom = 0.5

Another variation of a sequence based on the generating pattern s_i+1=a*s_i²+b.

This sequence quickly converges to a two cycle. 1.63, 0.37, … As you would expect the zoom-out picture look very similar to the two step sequence fractals studied back in June. See Sequence Fractals Part II #34 for example.

$ z^2+c+s_i$
s₀=0, s_i+1=-1.8*s_i²+0.9
Center:-0.5+0i; Zoom = 1

Today's generator creates a chaotic sequence of values between -0.58 and 0.90. There are no patterns, no repetition and no convergence to a cycle. Sequence Fractals Part III #27 also used a chaotic sequence. In the previous case the sequence was confined to the boundary of a circle. Here is sequence is contained in a closed segment of the real line.

$ z^2+c+s_i$
s₀=0.2, s_i+1=-1.8*s_i²+0.9
Center:-0.5+0i; Zoom = 1

Same setup as yesterday, Sequence Fractals Part III #34, but with a different start point for the sequence.

As is characteristic of chaotic sequences, a slightly different start point creates an entirely different sequence. And so the picture is very different.

$ z^2+c+s_i$
s₀=0.3, s_i+1=-1.8*s_i²+0.9
Center:-0.5+0i; Zoom = 1

Now the sequence start point is 0.3.

Changes to the start point make different pictures, but there are also some commonalities. They all look like colored Rorschach pictures. Since the sequence consists of all real numbers there is a symmetry between z and its conjugate $ \bar z$. That generates the vertical symmetry in the picture.

Also, nothing is captured. (The capture set is not colored black. It is colored blue/grey with this palette.) Well, maybe something is captured, I have not searched for a capture set. I suspect with the relatively large range [-0.58,0.90] for s_i, eventually every orbit gets bumped out to the escape region. When the sequence has a large bump, several points are knocked out together.

$ z^2+c+s_i$
s₀=0.4, s_i+1=-1.8*s_i²+0.9
Center:-0.5+0i; Zoom = 1

Now the sequence start point is 0.4.

$ z^2+c+s_i$
s₀=0.5, s_i+1=-1.8*s_i²+0.9
Center:-0.5+0i; Zoom = 1

One more, the sequence start point is 0.5.