Monday, October 23, 2017

Hobbymath's wanderings #19

Color wheel graphs of complex functions (Wiki).

Visualizing Complex Functions.

Conformal map visualizer (recommended!).

Benford's law (Wiki).

Domain coloring (Wiki).

Conformal Mapping.

Making stereograms with Python (recommended!).

Lifted Domain Coloring (pdf).

Domain coloring on the Riemann sphere.

Riemann sphere (Wiki).

Riemann Sphere.

Mercator projection (Wiki).

Azimuth (Wiki).

Plane-filling curves and fractals (pdf).

Automorphism (Wiki).

Rotational symmetry (Wiki).

Cyclic group (Wiki).

Root of unity (Wiki).

Root of unity modulo n (Wiki).

Creating Digital Chladni Patterns.

Python example: Chladni patterns.

Logarithmic derivative (Wiki).

Digamma_function (Wiki).

Polygamma function (Wiki).

Arc length (Wiki).

Lacunarity (Wiki).

Bifurcation theory (Wiki).

Random Attractors Found using Lyapunov Exponent.

About strange attractors.

Strange attractor identification.

Rössler attractor (Wiki).

About strange attractors.

Dynamical_system (Scholarpedia).

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: