#chaotic attractor

[thecornerofchaos]

I love these colours

This is a custom 4d Strange Attractor projected onto a 2d plane, rendered with my homemade attractor program.

I can (try to) do requests, within the scope of strange attractors - feel free to ask whenever.

[thecornerofchaos]

Hi! Can you share any video of your attractors?

They don't inherently move so I assume you either want a video of one being generated (not that interesting looking but I can do that if you'd like), or a video of one from changing perspectives (aka spinning, which I haven't implemented yet. I may reblog this with a spinning one when I get that working).

Seeing as I can't provide a video I'll at least provide a new image

[thecornerofchaos]

Funnily enough this colour scheme happened because of an error.

This is a custom 4d Strange Attractor projected onto a 2d plane, rendered with my homemade attractor program.

As usual, I can (try to) do requests, within the scope of strange attractors - feel free to ask whenever (and wherever).

[fractalotl]

"Figure 1: Left panel: Chaotic attractor of a driven anharmonic oscillator on the location-position plane of a stroboscopic map taken with the period of the driving. Right panel: Natural measure on the same chaotic attractor. Lighter colors indicate higher local values of the distribution. Both the attractor and the natural measure are fractal. (From T. Tel, M. Gruiz, Chaotic Dynamics, An Introduction Based on Classical Mechanics, Cambridge University Press, 2006, with permission.)"

Via Scholarpedia:

http://www.scholarpedia.org/article/Attractor_dimension

[iomikron]

The above images represent the attractor of Lorenz's system for successive values of one parameter. Keeping the other two parameters of the system fixed, we change the values of the last parameter and what we get is the 'period doubling route to chaos'; a very common transition scenario from periodic orbits to chaotic orbits. 

Begining from a simple solution represented by a very simple orbit, as the values of the parameter decreases, quite abruptly, emerges a new kind of 2-period orbit.

Decreasing further the value of the parameter after the 2-period orbits, 4-period orbits appear.

The third image represents the transition from 4-period to 8-period and, similarly, the fourth image represents the orbits of Lorenz system, for which the transition to 16-period orbits happens.

Generally, it can be proved that there are intervals in the range of parameter's values, for all the types of \(2^k\)-period orbits. For increasing values of k, corresponding to critical values of parameter, the width of these intervals decreases. Eventually, there is an asymptotic critical value of parameter c, for which the orbit becomes so dense that we get a chaotic attractor.