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Monday, October 23, 2017

Studying discrete-time dynamical systems (I): the double loop strange attractor

This is a little "experimental mathematics" study of the evolution of the strange attractors generated by a family of discrete-time dynamical systems, based on the following pattern:

x_n=sin(2 \cdot x_{n-1})+((1+(\frac{C}{40}))\cdot y_{n-1})

y_n=cos(x_{n-1})

The seed is (x,y)=(0,1).

The video shows the evolution of the family of strange attractors obtained after 10^6 iterations when C is an integer value increasing from 0 to 250.


Each frame of the animation is a specific strange attractor generated for a single value of C after 10^6 iterations. So the first frame is the strange attractor generated by C=0 after 10^6 iterations up to the last frame, generated by C=250 after 10^6 iterations.

The initial steps recall the movement of some thick smoke in the air (e.g. the smoke of a cigarette). I am calling it "The double loop strange attractor" because when C gets higher values it tends to have a very specific (kind of "art deco-esque") double loop pattern.

Update: It could be a "distant relative" of the De Jong strange attractor.

Update 2: This is the same family of strange attractors converting each (x,y) pair into polar coordinates:


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