Attracted to Attractors

What are Attractors

Attractors, in dynamical systems are structures that a lot of initial conditions evolve to over time. When the dynamics are chaotic, if an attractor exists, they often have a fractal structure which makes them gorgeous to visualize and are called strange attractors. Below are three, 3-dimensional strange attractors as well as the differential equations that generate them.

Thomas’ Cyclically Symmetric Attractor

Thomas’ Cyclically Symmetric attractor is governed by the following 3 differential equations

$\dot{x} = \sin(y) - bx$

$\dot{y} = \sin(z) - by$

$\dot{z} = \sin(x) - bz$

Rossler Attractor

The Rossler attractor is governed by the following 3 equations

$\dot{x} = - y -z$

$\dot{y}=x+ay$

$\dot{z}=b+z(x-c)$

Lorenz Attractor

The Lorenz attractor is governed by the following 3 equations

$\dot{x} = \sigma (y-x)$

$\dot{y} = x(\rho-z)-y$

$\dot{z} = xy - \beta z$

Code

The uncleaned up code can be found here.

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