$ 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=0.2, 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, 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=0.37, 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=-0.37, 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=0.0, 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=0.0, 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:-1.0314-0.0030i; 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:-0+0i; Zoom = 0.5

Here the sequence is generated much like orbits of the ubiquitous z^2+c fractals.

This looks similar to the quadratic case when you start someplace other than the critical point.

If we had seeded the sequence at exactly the fixed point (so that the sequence is a single repeated value) then we get a full Mandelbrot set, shifted -0.37. When we start anywhere else, the sequence bounces around a little before settling down. With that early bouncing, some orbits that would otherwise converge get knocked out to infinity.

There are also a few points, harder to identify, that would have escaped that get pushed into the capture set.

Where I live, this summer is on track to be the hottest summer ever. Now we are at the start of August, which has traditionally been the hottest month of the year. I am not enjoying the heat at all. However, I do enjoy generating fractals with the hot palette.

$ z^2+c+s_i$
s₀=0.4, s_i+1=(0.28+0.96i)*s_i
Center:-0.5677-0.5371i; Zoom = 10

This is one of my favorites. There is a lot going on.

The white area is a true capture area. Typically, with the non-convergent sequences, the bouncing around is enough to knock an otherwise convergent orbit into the escape region. Here, for some values of c, the basin of attraction is big enough that the orbits can be disturbed by a displacement of 0.4 without escaping. These orbits would not converge to a point or finite cycle, but they would remain bounded.

But, since some points are captured and some escape, there is a boundary between those two sets where interesting things happen. Typically, each iteration step removes a thin layer around the capture set, revealing a little more detail. There is a smooth color transition. That is happening here. In addition, the sequence gives the orbit an extra kick. That causes larger chunks to be kicked out to the escape set. You can see this in areas that look almost like a puzzle. Interconnecting pieces of distinctly different colors.

$ z^2+c+s_i$
s₀=0.4, s_i+1=(0.28+0.96i)*s_i
Center:0+0i; Zoom = 0.5

The multiplier is based on the 7-24-25 Pythagorean triple. The s_i bounce around the circle with radius 0.4 at approximately 73.7° increments.

$ z^2+c+s_i$
s₀=0.1, s_i+1=(0.6+0.8i)*s_i
Center:-0.6229+0.3970i; Zoom = 0.5

A zoom into the previous image Sequence Fractals Part III #24

$ z^2+c+s_i$
s₀=0.1, s_i+1=(0.6+0.8i)*s_i
Center:0+0i; Zoom = 0.5

The multiplier is based on the 3-4-5 Pythagorean triple. Note that the absolute value of the multiplier is 1. So, for all i, |s_i| = |s₀| = 0.1.

The angle is an irrational angle (cos^-1(0.8)). The sequence moves around a circle (radius = 0.1), without repeating any point, at approximately 53° each steps.

$ z^2+c+s_i$
s₀=1.0, s_i+1=-0.5*s_i
Center:-0.8172+0.185116i; Zoom =512

Another zoom. The multiplier in the sequence is negative. This sequence still converges to 0, but is bounces between positive and negative values along the way. The previous sequence took a direct path from 1 to 0.

$ z^2+c+s_i$
s₀=1.0, s_i+1=-0.5*s_i
Center-1.416536+-1.07538i; Zoom =8384

The first of two zooms.

The zooms are not much different from zooms for the quadratic case. As s_i+1 approaches 0, the iteration step get close to the usual $ z^2+c$ iteration.

$ z^2+c+s_i$
s₀=1.0, s_i+1=0.5*s_i
Center:0.0+0.0i; Zoom =0.5

Yesterday's sequence reset after 30 steps. Without a reset, arithmetic sequences head off to infinity. Fractal pictures with them just have some ghostly shadows, no chaos. The first few orbit steps for small numbers bounce around, but soon everything heads off to infinity. Only light shadows from the early iteration remain.

The same holds for geometric sequences that go to infinity. But not all geometric sequences blow up. Here is one that converges to 0. Usually the fractals built on these sequences, such as this one, have a large capture set, and no need to reset after a few steps.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 30 steps.
Center:0.0+0.0i; Zoom =0.5

At 30 steps between resets the white capture set is gone.

I found some cracks that host minis, similar to yesterday's post. Sequence Fractals Part III #19 Much smaller than yesterday's, but after adjusting the scale they were so similar they do not deserve a separate post.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 25 steps.
Center:-1.2101-0.41833i; Zoom = 32

Found a mini in a crack.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 25 steps.
Center:0.0+0.0i; Zoom =0.5

At 25 steps between resets there is a small capture area in the nose.

One zoom coming up for this one, a peek at a brown spot in the wings.

$ z^2+c+s_i$
s₀=0.0, s_i+1=s_i+0.1. Reset after 20 steps.
Center:-0.98714+0.296305i; Zoom = 2048

Another zoom. The color entropy dial is turned down a little.