**Slope Formula Tutorial for UltraFractal - Page
7**

**Strange Attractors**

In 1961 Edward Lorenz, a meteorologist at MIT, discovered what is now called the Butterfly Effect while simulating weather on his computer. He found that some iterative equations could produce dramatically different results with extremely small changes in the starting parameters. His most well known set of equations is the Lorenz Attractor. It is a model of turbulent flow.

dx/dt = a(y - x)

dy/dt = bx - y - zx

dz/dt = xy - cz

where x, y and z are variables, and a, b and c are constants. The time sequence for x, y and z gives orbits that spiral around each other, but in an unpredictable manner. The orbits show the behavior of fractals.

The most interesting strange attractors, from an artistic point of view, are the 3D and 4D attractors. The 2D attractors can also produce some interesting patterns. The slope algorithm can produce very nice 3D effects, and can even be animated with Ultra Fractal's animation capability. The standard parameters for the slope strange attractors were presented in the Introduction. The use may want to review those parameters before continuing. Reb.ufm contains 12 slope strange attractor formulas. Nine of the formulas are based upon the work of Julian Sprott, who studied various power series, ordinary differential equations using power series, and sin/cos series as a source of strange attractors. Each of these formulas has a large number of preset options for generating different strange attractors.

The first set of examples uses the Slope 4D Quadratic ODE G
Attractor with select preset coeffs = 9. This is one of the Sprott attractors,
with the attractors being solutions to quadratic ordinary differential equations
in 4 dimensions. One of the characteristics of many strange attractors is that
with a sufficiently large hit density a seemingly continuous smooth surface is
produced. At lower hit densities it can be seen that the probability of finding
an orbit is quite variable, giving an interesting structure to the attractor.
Sprott4DQ and Sprott4DQrot have two layers of the strange attractor with one
layer having a hit density of 5 (tan) and the other with a hit density of 0.1
(orange). The tan coloring uses the tan preset in Direct Color Slope while the
orange coloring uses the yellow preset. The orange layer is on top with a merge
mode of "saturation".The second image (Sprott4DQrot) has been rotated
60^{o} around the axis perpendicular to the XW plane and -160^{o}
around the axis perpendicular to the ZW plane. This set of rotations moves a
portion of the attractor that was in the 4th dimension into the 3D space that we
can observe. Adjustments need to be made to the rotation centers and final
translation amounts to keep the rotation version centered. The user should
experiment with the parameters to find the correct setting on their own.

Sprott4DQ rot center X = 0.8288 rot center Y = 1.12772 rot center Z = 1.589674 rot center W = 6.7663 final Z = -0.5 |
Sprott4DQrot rot center X = 0.8288 rot center Y = 1.12772 rot center Z = 1.589674 rot center W = 6.7663 final X = 0.23098 final Y = -0.21739 final Z = -0.5 |

Some strange attractors can be quite lacy, as is shown in Sprott3DC, which uses the Slope 3D Cubic ODE G Attractor with select preset coeffs = 3. The Pickover Attractor has long been one of my favorites. The image GenPick uses the Slope Pickover G Attractor, and is a three layer image. The middle layer has Hit Value = Largest Z, with Hit Density = 1, Filter Width = 2 and Orbit Closeness = 3. Direct Color Slope has a preset of MediumOrchid. This gives a smooth, shiny image looking somewhat like pulled taffy. The top layer adds some texture with Hit Value = Count, Filter Width = 1 and Orbit Closeness = 1. Direct Color Slope has a preset of Coral. The layer is merged using "Multiply" The user should experiment with background layer on their one. The one shown comes from Manowar Julia as the ufm and Attractor Traps as the ucl with the Pickover attractor selected.

Sprott3DC X rotation = 32.3 ^{o
}rot center X = -0.27174rot center Y = 0.57065 final X = 0.24457 |
GenPick Y rotation = 40 ^{o
}parameter b = 1.17728parameter e = 0.6875 |

This tutorial section isn't complete without an example for a 2D attractor. Latoo was created with the Slope Latoocarfian G Attractor. It is a three layer image with the top layer colored with 3D Texturizer Enhanced III in its non-slope coloring mode and the middle layer is colored with Direct Color Slope.

Latoo |