Hobbymaths: October 2017

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:

Hobbymaths

Monday, October 23, 2017

Hobbymath's wanderings #19

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