### Systems of two real variables

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• 3f
• Posts: 1777

#### Re: Systems of two real variables

« Reply #15 on: June 12, 2019, 06:59:14 PM »
I've played with pure quadratics (only x^2 , y^2, and xy). You have 6 parameters but with a linear "conjugation" you can bring it into canonical form with just two; Claude implemented it in Kalles Fractaler not too long ago.
Canonical form (c is pixel):
$$x \leftarrow x^2 \pm y^2 +c_x\\ y \leftarrow ax^2 +2bxy + c_y$$
It has inside "minis" but they are all different shapes. Example is a=3, b=-3 with "+" sign rendered with fairly normal iteration coloring. Black is really inside.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Systems of two real variables

« Reply #16 on: June 12, 2019, 11:25:51 PM »
Paul, have you generated and x/y images for the Discrete Volterra-Lotka?

Peitgen and Richter devote a short chapter to these equations in The Beauty of Fractals. (A thirty three year old book, but still the most referenced book on my bookshelf.) Is this your source? They have eight images, four with h=0.739, p=0.739. and four with h=0.8, p=0.86. The have one, very poor, probably hand drawing two tone image of the h-p plane. Normal coloring should work for the x/y plane. The dynamics have one or two attraction basins, plus escape-to-infinity.
Earl Hinrichs, offering free opinions on everything.

#### Spyke

• Strange Attractor
• Posts: 98

#### Re: Systems of two real variables

« Reply #17 on: June 12, 2019, 11:58:48 PM »
According to Peitgen and Richter, in the area in the upper right, the fixed point (1,1) is attractive. Moving left and down, it bifurcates and becomes an attractive invariant circle. This would the teeth, if you are calling the darker regions the teeth, plus perhaps (continuing the analogy) some "gum" in the transition. P&R call the lighter area between the teeth, tongues. I guess they are putting the head on the other side. Anyway, that area has an attractive, finite, periodic orbit, and it is called resonance. Beyond that you get multiple disjoint attracting orbits or orbits and circles, then it transitions to strange attractors, and then escape to infinity.

I kept confusing myself as I typed the about. I had to stop and think about which plane is which. Paul's picture is of the h/p plane (M-like). The dynamical behavior is the dynamics on the x/y plane (J-like), with h & p parameters.

• 3f
• Posts: 1777

#### Re: Systems of two real variables

« Reply #18 on: June 13, 2019, 02:14:16 AM »
Embedded Julia set in
$$x \leftarrow x^2 - y^2 +c_x\\ y \leftarrow ax^2 +2bxy + c_y$$
with a=-0.571431, b = 0.886901. Those black minis are smooth (not fractal).
I think the inherent skew, like in the Henon images, gives it a bit of a 3D look.
« Last Edit: June 13, 2019, 02:26:39 AM by gerrit »

• 3f
• Posts: 1366

#### Re: Systems of two real variables

« Reply #19 on: June 13, 2019, 02:58:21 AM »
There are several Volterra-Lotka Julia fractals in my gallery:

https://fractalforums.org/index.php?action=gallery;su=user;cat=43;u=97

The six images starting with "Superstring Theory" are Volterra-Lotka. The named one has a strange attractor and "Neon Downtown Alpha Centauri" has an invariant circle, or limit circle; both highlighted in neon blue. The bright spots in "The Pleiades Vortex" are a period 7 discrete cyclic attractor. The paler brown haze in the "Smelter" is a strange attractor. "Organic Twist" has holes in the basin of a fixed point; in this case, points in the holes escape, but there are parameters where they'd go to a secondary cyclic attractor instead.

• 3f
• Posts: 1777

#### Re: Systems of two real variables

« Reply #20 on: June 14, 2019, 02:57:06 AM »
Fighting minis catching the wave.
$$x \leftarrow x^2 - y^2 +c_x\\ y \leftarrow ax^2 +2bxy + c_y$$
with a=-1.38095, b = -0.04762.

• 3f
• Posts: 1366

#### Re: Systems of two real variables

« Reply #21 on: June 14, 2019, 05:07:52 AM »
Have you tried using orbit colorings inside of those black objects, gerrit? Try any of: triangle inequality average, elliptic harlequin, smooth traps for starters.

• 3f
• Posts: 1777

#### Re: Systems of two real variables

« Reply #22 on: June 14, 2019, 08:15:16 AM »
Have you tried using orbit colorings inside of those black objects, gerrit? Try any of: triangle inequality average, elliptic harlequin, smooth traps for starters.
Like Mandelbrot minis that produces nothing interesting, for my taste.

• 3f
• Posts: 1366

#### Re: Systems of two real variables

« Reply #23 on: June 14, 2019, 11:48:19 AM »
The Volterra-Lotka parameter object, some more.

Image 1: A closeup in a similar location to the last two, this time combining |turning 2| and y averages over the orbit and squaring the former average. It's remarkable how the results of viewing this parameter object with these methods produces images resembling clouds, fire, and other such non-solid phenomena.

Image 2: One of the "teeth". The maroon area at top left is periodic. The red band adjacent to it has strange attractors and islands of order. The yellow "tooth" is a bundle of narrow periodic regions and regions producing limit circles. Coloring averages x * |turning 2| and |x| * y. The average from the former is again squared.

Image 3: The next "tooth" up from the one in image 2. A similar pattern occurs, but there is a very prominent "swallow" island in the chaos, and this one's coloring makes clear that the "tooth" and the triangular periodic region overlap partially, producing a region where the Julia sets have two distinct finite attractors. Same coloring method as image 2 but with a new color gradient.

Image 4: A region closer to the origin, with higher periods. |turning 2|(x2 + y2) averaged and squared for one dimension of the coloring, and |x| * y gives the second.

• 3f
• Posts: 1366

#### Re: Systems of two real variables

« Reply #24 on: June 16, 2019, 03:06:45 PM »
More Volterra-Lotka parameter object images.

1: Smooth Traps coloring combining a rotated half-plane passing through (1, 1) and the outside of the disk inscribed in the square from 0 to 2 along both axes. A chaotic region from one of the earlier images is shown again.

2: Same coloring method, of one of the "teeth".

3: A band of chaos adjacent to a "tooth". Image has been rotated. Colored by a ribbon trap of width 2 whose midline goes through (1, 1) and a radius-3 disk interior with the same center point. (Remember, this point is always a fixed point in the Volterra-Lotka dynamics, so it makes a good smooth-traps center point.) Note the "swallows" familiar from forced logistic and Henon parameter spaces turning up again here quite clearly.

4: A "tooth" colored by a combination of the same radius-3 smooth trap and "velocity over magnitude". That method averages, over the orbit, $$\frac{|P_n - P_{n - 1}|}{|P_n|}$$ where || is the Euclidean norm and the orbit points P are generic vectors in a generic inner-product space. As such this method is applicable to a very broad set of fractals -- in particular it should work on Mandelbulb, Mandelbox, and other more-than-two-dimensional ones as well.

All 12 of the recent Volterra-Lotka images posted to this thread use the following sampling method: for each pixel, 100 parameter samples are selected from inside that pixel in a grid pattern, and for each parameter sample, 1 random dynamic-plane starting point is selected from a rectangular region (usually the square from 0 to 1 or from 0 to 2 on each axis). The 100 orbits produced from each parameter/dynamic sample pair each get the two coloring-index values calculated by the two coloring methods combined for that image; the two colors for a single parameter sample/orbit are combined by HSL addition; and the 100 resulting combined colors for all the parameter samples from a pixel are averaged to color that pixel.

• 3f
• Posts: 1366

#### Re: Systems of two real variables

« Reply #25 on: June 18, 2019, 01:55:18 AM »
Another system of note is Ibiza:

$$x_{n+1} = 1 - ax_n^2 + y_n$$
$$y_{n+1} = bx_n y_n + cy_n^3 - 1$$

As you can see it's a modified Hénon map, with a more complex function for y.

Image 1 is a sample Julia set. Escaping points are yellow, colored by smoothed iterations. The dark basin has a finite attractor, and it is strange (purple thread with magenta halo). As you can see, this one resembles a Volterra-Lotka attractor more than it does a Hénon one.

The next images are parameter space images in the a-b plane for fixed c. First two of these has c = 0.1 and use orbit-average of x2 + y2 combined with the barycenter of the atan()s of the attractor points. The second is a zoom of the first. Third has c = 0.2 and colors by |turning 2| and plain attractor barycenter (no atan).

The "mountains" have long "glaciers" where periodic cycles alternate with limit circles, much like the fringe zone of the Volterra-Lotka parameter space. But the sides and peak of the "mountains" tend to be rich in strange attractors, periodic islands, and other critters.

• 3f
• Posts: 1366

#### Re: Systems of two real variables

« Reply #26 on: June 28, 2019, 06:37:56 AM »
Image 1: Ibiza parameter ab plane for c = 0.2. Colored by a combination of orbit averaged |turning 2| and a smooth traps disk about (0.02381,-0.70238). Magnified from previous.

Image 2: Same plane, location a bit to the right of previous, colored by a combination of triangle inequality average and attractor barycenter x. The latter skips every 2nd point in the attractor.

Image 3: Zoom of previous image near top center, but colored using the method used for the first of these three images, but with a new gradient.

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