Rössler System

The Rössler attractor, a simple system with spiral chaos.

The Rössler system is an artificially constructed system designed to behave similarly to the Lorenz system but with only a single non-linear term. It is widely used to study chaos and topological properties of attractors.

Equations

\begin{align} \dot{x} &= -y - z \\ \dot{y} &= x + a y \\ \dot{z} &= b + z (x - c) \end{align}

Parameters

The standard parameters producing the classic Rössler attractor:

Parameter	Symbol	Default Value	Description
`a`	$a$	$0.2$	Fold parameter
`b`	$b$	$0.2$	Injection parameter
`c`	$c$	$5.7$	Dissipation parameter

Equilibria

The Rössler system has two equilibria if $c^2 > 4ab$ :

\begin{align} O_1 &= \left( \frac{c - \sqrt{c^2 - 4ab}}{2}, \frac{-c + \sqrt{c^2 - 4ab}}{2a}, \frac{c - \sqrt{c^2 - 4ab}}{2a} \right) \\ O_2 &= \left( \frac{c + \sqrt{c^2 - 4ab}}{2}, \frac{-c - \sqrt{c^2 - 4ab}}{2a}, \frac{c + \sqrt{c^2 - 4ab}}{2a} \right) \end{align}

Usage

from hidden_attractors import get_system

rossler = get_system("rossler")

params = rossler.default_params()
eq = rossler.equilibria_fn(params)