## A Million Dots

# Dynamics & Strange Attractors

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An attractor is a set of states for a dynamical system towards which arbitrary states tend to evolve; attractors are called ‘strange’ when they exhibit fractal-like behavior.

This program visualizes several three dimensional dynamical systems with strange attractors, accessible via the menu in the top right. The defining differential equations are below:

### Aizawa

The aizawa attractor arises from the following dynamical system with parameters $a=0.95$, $b=0.7$, $c=0.6$, $d=3.5$, $e=0.25$ and $f=0.1$. $$ x^\prime = (z-b)x-dy $$ $$ y^\prime = dx+(z-b)y $$ $$ z^\prime = c+az-\frac{z^3}{3}-(x^2+y^2)(1+ez)+fzx^3 $$

### Chen

The Chen attractor arises from the following dynamical system with parameters $a=5$, $b=-10$, and $d=0.38$.

$$ x^\prime = ax-yz $$ $$ y^\prime = by+xz $$ $$ z^\prime = dz+xy/3 $$

### Dadras

The Dadras attractor arises from the following dynamical system with parameters $a=3$, $b=2.7$, $c=1.7$, $d=2$ and $e=9$. $$ x^\prime = ax-yz $$ $$ y^\prime = by+xz $$ $$ z^\prime = dz+xy/3 $$

### Rossler

The Rossler attractor arises from the following dynamical system with parameters $a=b=0.2$ and $c=5.7$.

$$ x^\prime = -(y+z) $$ $$ y^\prime = x+ay $$ $$ z^\prime = b+z(x-c) $$

### Sprott

The Sprott attractor arises from the following dynamical system with parameters $a=2.07$ and $b=1.79$.

$$ x^\prime = y+axy+z $$ $$ y^\prime = -bx^2+yz $$ $$ z^\prime = x-(x^2+y^2) $$

### Thomas

The Thomas attractor arises from the following dynamical system with parameter $b=0.208186$ $$ x^\prime = \sin(y)-bx $$ $$ y^\prime = \sin(z)-by $$ $$ z^\prime = \sin(x)-bz $$

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