$ 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.